MORSE THEORY ON HILBERT ~ANrF~LDS - Richard Palaisvmm.math.uci.edu/PalaisPapers/MorseThOnHilbertManifolds.pdf · MORSE THEORY ON HILBERT ~ANrF~LDS ... M,) in terms of the critical

romiogY Vol. 2, pp. 299-340. Pcrgamon Press. 1963. Printed in Great Brimin

MORSE THEORY ON HILBERT ~ANrF~LDS

RICHARD S. PALAIS

THE TERM “MORSE THEORY' is usually understood to apply to two analagous but quite distinct

bodies of mathematical theorems. On the one hand one considers a smooth, real valued

functionfon a compact manifold M, defines M, = f - ‘[ - x), a], and given a closed interval

[a, 61 describes the homology, homotopy, homeomorphism or diffeomorphism type ofthe pair

(Mb, M,) in terms of the critical point structure off in f-‘[a, b]. On the other hand one

takes a compact Riemannian manifold V, defines M to be the ‘loop space’ of piecewise

smooth curves joining two points (with a natural topology) and f: N -+ R the length

function and again describes the homology and homotopy type of (Mb, M,) in terms of the

critical point structure offinf-‘[a, 61 (i.e. in terms of the geodesics joining the two points

whose Iengths lie between a and b). The classical approach to this second Morse theory is

to reduce it to the first Morse theory by approximating M, (up to homotopy type) by

certain compact submanifolds of piecewise broken geodesics. A particularly eIegant and

lucid exposition of this cfassical approach can be found in John Milnor’s recent Annals

Sri&y [S].

Our goal in the present paper is to present a Morse theory for differentiable real

valued functions on Hilbert manifolds. This encompasses both forms of Morse theory

mentioned above in a unified way. In addition the generalization of the Morse theory of

geodesics to higher loop spaces (i.e. maps of an n-disk into a manifold with fixed boundary

conditions) and even more general situations works smoothly in th.is framework, whereas

previous attempts at such generalizations were thwarted by the lack of a good analogue of

the approximating compact manifolds of piecewise broken geodesics.

We have endeavored to make the exposition relatively self contained. Thus the first

two sections give a brief resume of the classical theory of Frechet on the differential calculus

of maps between Banach spaces (details and proofs will be found in [I]) and in sections

3 to 9 we give a brief treatment of the theory of Banach manifolds with particular emphasis

on Hilbert manifolds (details and proofs will be found in [4]).

In sections 10 through 12 we prove the

MAIN THEOREM OF MORSE THEORY

Let M be a complete Riemannian manifold of class C.“’ (k > I) and f: M + R

a C”“‘-function. Assume that all the critical points offare non-degenerate and in addition

300 RICHARD S. PALAIS

(C) If S is any subset of Mon whichf is bounded but on which /I Vfll is not bounded

away from zero then there is a critical point off adherent to S. Then

(a) The critical values off are isolated and there are only a finite number of critical

points off on any critical level;

(b) If there are no critical values offin [a, b] then Mb is diffeomorphic to M,,;

(c) If c c c < b and c is the only critical value offin [a, 61 and pi, . . ., p, are the critical

points off on the level c, then Mb is diffeomorphic to M, with r handles of type (k,, I,),

. ..( (k,, I,) disjointly CL-attached, where ki and li are respectively the index and co-index

Of pi.

It should be noted that iffis proper, i.e. iff-‘[a. 61 is compact for every closed interval

[u, b], then condition (C) is automatically satisfied, hence the Morse theory for compact

manifolds is included in the above theorem. On the other hand if M is infinite dimensional,

hence not locally compact, then it is impossible for a real-valued function f: M + R to

be proper whereas we shall see condition (C) is still satisfied in cases of significant interest.

A theorem similar to the above was obtained independently and essentially simul-

taneously by S. Smale.

In $13 we show how to interpret the loop space of a complete finite dimensional

Riemannian manifold V as a complete infinite dimensional Riemannian manifold M.

This is due to Eells 121, however we have followed an approach suggested by Smale. In

$14 we show that if we take f: M + R to be the ‘action integral’ then the hypotheses of

the Main Theorem are satisfied, thereby deriving the Morse theory of geodesics. In $15

we return to the abstract Morse theory of functions satisfying condition (C) on a Riemannian

manifold and in particular derive the Morse inequalities. Finally in $16 we comment

briefly on generalizing the Morse Theory of geodesics to higher loop spaces, a subject we

hope to treat in detail in a later paper.

41. DIFFERElVl-LUILITY

Let V and W be Banach spaces, 8 an open set in V and f: 0 -+ W a function. If

p E 0 we say that f is differentiable at p if there exists a bounded linear transformation

T: V + W such that /f(p + x) -f(p) - TX ]l/]l )] x - 0 as x -+ 0. It is easily seen that

T is uniquely determined and it is called the differential off at p. denoted by dfp. The

following facts are elementary [l, Ch. VIII]:

Iff is differentiable at p then f is continuous at p;

Iffis differentiable at p and U E 8 is a neighborhood ofp then g = f 1 U is differentiable

at p and dgp = df, ;

Iffis constant then it is differentiable at p and d& = 0;

If S : V --* W is a bounded linear transformation and _f = S]8 then f is differentiable

at p and df, = S;

If f is differentiable at p, g : 0 + W is differentiable at p and z, /I are real numbers

then (zf + pg) is differentiable it p and d (zf + /?g), = adf, + /?dg,;

MORSETHEORY OK WLBERT MA~-lFOLDS 301

If U is a neighborhood off@) and g : U + Z is differentiable at_@) then iffis differentiable at p, g qf is differentiable at p and d(g 3f)p = dgft,, D df,.

Now suppose f is differentiable at each point of 8. Then df: p -+ d& is a function from 8 into the Banach space L(V, W) of bounded linear transformations of Y into W (sup. norm). If df is continucrus then we say thatjis of class C’ in 0. If dfis differentiable at a point p E B then d(df), = d’jfp E L(K L(V, W)). We make the usual canonical identification of L(V, L(V, W)) with L2(Vs W), the space of continuous bilinear maps of V x V into W. Thus d2fP is interpreted as a bilinear map of V x V into W and it can easily be shown to be symmetric ]l, p. 1751. In case d’& exists at each p E 0 and the map d2f: p -f d2fP is continuous we say that f is of class C2 in 0. Inductively suppose dkf: 0 I* S’(V, W) exists and is differentiable at p. Then (d’” tf>, = d(dy), E

L( V, tk( V, W)) z Lk+‘( V, W) and dkflfp. the (k + I)st differential offat pY is a bounded, symmetric [ 1, p, 1761 (k + 1).linear map of V x .,. x V into W. If d”ifP exists at each pointpEIIanddk”‘f:@-+L k’ ‘( V W) is continuous then we say that f is of class Ckfl in 0. Iffis of class C’ in 0 for ever; positive integer k we say thatfis of class C” in 0.

A linear map of R into a Banach space W is completely determined by its value on the basis element 1. We use this fact to define the derivative of a differentiable function f‘: 19 -+ W where 0 2 R: namely the derivative off at p, denoted by J’(p)+ is defined by f’(p) = df,(l)% so d&(a) = af’(p). If f is differentiable at each point of 0 we have f ’ : U -+ W, and if f is of class C’ in 0 we can define f ” = (f ‘)’ and in generai if f is of class CL we can define f (kf : Q --t W. Clearly the relation off”’ and dLfis dk&(,<t, . . . . XJ = xix2 . . . M”(*YP). If g : W + Z is differentiable then (g Olf)‘tpj = d(g Qf),(I) =

dg~~~~(d~(l~~ = d~~,~~~ff~~~, i.e. (9 of)' = dsI cf'. For future reference we note the following. If B is a continuous symmetric bilinear map

of Y x V into FV thenf: Y + W defined byfi:n) = B(G’, v) is of class C”. In fact d&(u) I = 2B(p, mu). d’f = ZB and d’f = 0.

,_ . . St. THREE BASIC THEOREMS

Let V and W be Banach spaces, p E V, 0 a convex neighborhood of p in V andf : Q -+ W a Ck-map, k r 1. Then there is a CL-‘-map R, : S-+L(V, W).such that 1yx =p + ve8

f(x) =f(p) + R,(x)c.

COROLLARY (TAYLOR'S THEOREM). If m I k there is a Ck-mwmap R, :cI1 + Lm( V, W) such that ifx = p + v E: 0 then

f(x) =/(P) + d&.(u) + + d2f,(u, 01 + . . . + (m ; ,) ! d”- ‘fp(u, 3. a v) + R,(x)(v, .., u).

La V and W be ~li&ueh spaces, 0 open in V, and f : 0 + W a Cc-map, k L I. Let p E B and suppose that dfF mops V me-m-me mm W. Then there is a neighborhood U of


p included in B such that f 1 U is a one-to-one map of U onto a neighborhood of f(p) and more-

over (f I V)- I is a Cc-map off(U) onto U.

DEFINITION. If 8 is an open set in a Banach space V then a C’-vector jietd in 0 is a

C’map X: B + V. A solution curve X is a Cl-map o of an open interval (a, b) E R into r?

such that o’ = X 0 CT. If 0 E (a, b) we call a(O) the initial condition of the sohaion a.

The following theorem is usually referred to as the local existence and uniqueness

theorem for ordinary differential equations (or vector fields).

THEOREM. Let X be a CL-vector field on an open set 0 in a Banach space V, k 2 1.

Given p. E 0 there is a neighborhood U of p. included in 0, an E > 0, and a CL-map

q : U x (- E, E) --) V such that:

(1) Ifp E U then the map op : (-E, E) + V defined by a,(t) = cp@, t) is a solution of

X with initial condition p:

(2) If u : (a, b) --* V is a solution curve of X with initial condition p E U then a(r) =

a,(t) for It] < E.

The proofs of these three basic theorems can all be found in [l] or in [4].

$3. DIFFEREF44LX MANIFOLDS WITH BOUNDARY

If k is a bounded linear functional on a Banach space V, k # 0 we call H=

{v E V(k(v) 1 0) the (positive) half space determined by k, and dH = {v E V(k(v) = O> is

called the boundary of H. A function f mapping an open set 0 of H into a Banach space U’

is said to be of class CL at a point p E 0 n dH if there exists a C’-map g : U -+ W, where U

is a neighborhood ofp in V, such that f (0 n U = gj8 n V. It is easily seen that d”f, = dmg,

is then well defined for m I k and that iffis of class C’ at each point of 0 n 8H and also in

0 - 8H then d*f: 0 + L”(V, W) is continuous for m 5 k; in this case we say that f is of

class Ck in 0. Next suppose that f is a one-to-one Ck-map of 0 into an open half space

K in W. We say that f is a Ck-isomorphism of 0 into K if f(S) is open in K and iff -i is of

class CL (if k 2 1 then it follows from the inverse function theorem that this will be so if

and only if dfP maps V one-to-one onto W for each p E 0).

Invariance of Boundary Theorem

Let H, be a half space in a Banach space V, and 0, an open set in H, (i = 1,2). Let

f : 0, + O2 be a C’-isomorphism. Then tf- either k 2 1 or dim Vi < co f maps 8, n aH,

Ck-isomorphically onto O2 n dH,.

Proof. In case k 2 1 the result is an immediate consequence of the inverse function

theorem. In case dim Vi < cg the theorem follows from invariance of domain.

Caution. In case k = 0 and dim V, = 00 the conclusion of the theorem may well fail.

Thus if V is an infinite dimensional Hilbert space and H a half space in V it is a (non-

trivial) theorem that H and H - dH are homeomorphic.

MORSE THEORY ON HILBERT MANIFOLDS 303

In the following, to avoid logical difficulties, we shall fix some set S of Banach spaces

and whenever we say Banach space we will mean one which belongs to the set S.

A chart in a topological space X is a homeomorphism of an open set D(cp) of X onto

either an open set in a Banach space or else onto an open set in a half space of a Banach

space. Two charts p and $ in X. with U = D(q) n D(G). are called CL related if $ O cp-’

is a Ck-isomorphism of q(U) onto t,b(U). A C’-atlas for X is a set ri of pairuise C’-related

charts for X whose domains cover X, and A is called complete if it is not included in any

properly larger CL-atlas for X. It is an easy lemma that if each of two charts cp, $ in X

is Ck-related to every chart in A then cp and li/ are CL-related. It follows that there is a

unique complete C’-atlas 1 including A. namely the set of all charts cp in X such that 43

is Ck-related to every chart in A, A’ is called the completion of A.

A CL-manifold with boundary is a pair (X, A) where X is a paracompact Hausdorff

space and A is a complete CL-atlas for X. In general we will use a single symbol, such as

.M. to denote both a C’-manifold (X, A) and its underlying topological space X, and

elements of A will be referred to as charts for M. If p E M a chart at p is a chart for M

having p in its domain. If A is a (not necessarily complete) CL-atlas for X then by the

C’-manifold determined by A we mean the pair M = (X, 2). If m < k then A is a Cm-atlas

for X and so determines a Cm-manifold which we also denote by ,ti (an abuse of notation),

so that a Ck-manifold is regarded as a Cm manifold if m I k.

If M is a C’-manifold, k 2 1, we define dA4 to be the set of p E A4 such that there exists

a chart cp at p mapping D(q) onto an open set in a half space H so that p(p) E dH. It

follows from the invariance of boundary theorem that every chart at p has this property

and also that {cp[aM}, where cp runs over the charts for M, is a C’-atlas for iiM, so dM is

a C*-manifold. Moreover we have the obvious, but satisfying relation ;I(aM) = 4,

If M and N are C’-manifolds a functionf: IV -+ N is said to be of class CL near p if

there exists a chart cp at p and a chart 1(1 at&) such that (i/ Of 5 9-l is of class CL, and fis

said to be of class C’ if the latter holds for each p E M. It is easily seen that f: ,U_ + N is of class C” if and only if $ 3 f 3 cp -' is C’ for every chart cp for iii and $ for N.

If we define objects to be C’-manifolds and morphisms to be P-maps then the axioms

for a category are satisfied.

84. TANGEh-T SPACES AND DIFFERENTIALS OF IMAPS

Let {Vi];;, be an indexed collection of Banach spaces and for each (ij) E Z* let cpij

be an isomorphism of V’j with Vi (as topological vector spaces) such that cpii = identity

and cpuqjk = ~ik. From the data {Vi, vii} we construct a new Banach space Y (by a

process we shall call amalgamation) and a canonical isomorphism ILi : V -+ Vi such that

rri = pijnj. Namely V is the set of {oi} in the Cartesian product of the Vi such that

Ui = VijUj. Clearly V is a subspace of the full Cartesian product, hence a topological vector

space. We define rri to be the restriction of the natural projection of the Cartesian product

onto V,. To prove that V is a Banach space and n, an isomorphism it suffices to note that

there is an obvious continuous, linear. two sided inverse 1, to rrj, namely Ij(v)i = cp,,(u).

304 RICHARD S. PALMS

Given a second set of data {IV,, +t,} satisfying the same conditions (with indexing

set K) suppose that for each (i, k) E I x K we have a bounded linear transformation

Tki : Vi + W’k such that \I/~ITri’pij = Tk,. Then if W is the amalgamation of {W,, +,..I

there is a uniquely determined bounded linear map T: V -, W such that n,T = T,JK~.

namely if {vi} E V then T {ci} = {We} where tipt = Tkici. T is called the amalgamation of

the T,,.

Now suppose M is a C’-manifold k 2 1, p E M and let I be the set of charts at p.

Given cp E I let V,,, be the target of cp (i.e. the Banach space into which cp maps D(p)). Then

for each (cp, $) E 1’ d($ O v-‘)~~~) is an isomorphism of I’,+, onto V,. Clearly d(ip O (P-‘)~,~)

= identity and by the chain rule d(f. d (p-1)0(p) O d(rp O $-l)+(,,) = d(l ., +-‘)40,,. Hence

the conditions for an amalgamation are satisfied. The resulting amalgamation is called

the tangent space to M at p and denoted by M,.

Let N be a second C’-manifold f: M + N a C’-map and K the set of charts at f(p). For each (cp, $) E I x K we have a linear map d($ Of o 'p-')+,(,,) of V, into W+. Moreover

the abstract condition for amalgamating is clearly satisfied, hence we have a well determined

amalgamated map dfP : M, 4 Nftpj called the differential off at p,

$5. THE TANGENT BUNDLE

Let n : E -+ B be a CL map of Ck-manifolds and suppose for each b E B n-‘(b) = Fb

has the structure of a Banach space. We call the triple (E, B, x) a Ck-Bunach space bundle

if for each b,, E B there is an open neighborhood U of b. in B and a C’-isomorphism

f:Ux Fbozn - ‘(U) such that u + f(b, u) is a linear isomorphism of F,, onto Fb for each

b E U. If (E’, B’, n’) is a second C’-Banach space bundle then a CL-map f: E + E’ is

called a C’-bundle map if for each b E B f maps Fb linearly into a fiber FYfb,. The map

f : B -+ B’ is then CL and is called the map induced by J

Let M be a CL+ l-manifold with boundary. Let T(M) = ,yM M, and define

II : T(M) + M by n(M,) = p. Given a chart cp for M with domain U and target V,

define cp : U x V, -+ n- '(U) by letting u -+ cp(p, v) be the natural isomorphism of I’,

with M,. Then it is a straightforward exercise to show that the set of such cp is a CL-atlas

for a CL-manifold with underlying set T(M) and moreover that T(M) is a Ck-Banach space

bundle over M with projection II. Iff is a C’+ ‘-map of M into a second C’+ i-manifold N

we define df: T(M) -* T(N) by df lA4, = dfP. Then one shows that df is of class Ck and

is a bundle map which clearly has f as its induced map.

The category whose objects are C’-Banach space bundles and whose morphisms are

CL-bundle maps is called the category of CL-Banach space bundles. The function M + T(M),

f --* df is then a functor from the category of Ck+ ’ -manifolds with boundary to the category

of C’-Banach space bundles. Since each author has his own definition of the tangent

bundle functor it is useful to have a general theorem which proves they are aLl naturally

equivalent, i.e. a characterization of T up to natural equivalence in purely functorial terms.

To this end we first note two facts. If 0 is an open subset of a CL-manifold M then 0 is in

MORSE THEORY ON HILBERT MASIFOLDS 305

a natural way a C’-manifold called an open submanifold of M: namely a chart for 0 is a

chart for M whose domain is included in G. If M is a 3anach space V or else a half space

in a Banach space V then the identity map of M is a chart in M and its unit class is a C” r- atlas for M. The Ckfl-manifold defined by this atlas will also be denoted by M. The

corresponding full subcategory of the category of Ckf ’ -manifolds with boundary which

we get in this way will be referred to as the subcategory of Banach spaces and half spaces.

On this subcategory we have an obvious functor T into the category of Ck-Banach space

bundles; namely with each such CL+’ -manifold M we associate the product bundle

r(M) = M x V considered as a Ck-Banach space bundle, and iff: M -+ N is a CL+ ‘-map.

where N is either a Banach space W or else a half space in W, then the induced map

r(f) : M x V --* N x W is given by r(f)(m, I’) = (f(m), dfm(r)). We now characterize

the notion of a tangent bundle functor.

DEFINITION. A functor t from the category of C”’ -manifolds to the category of C”-

Banach space bundles is called a tangent bundle functor if

(1) t(M) is a bundle over M and tff : M -+ N then f is the induced map of t( f ),

(2) Restricted to the subcategory of Banach spaces and halfspaces t is naturally equivalent

to 5,

(3) If M is a C’+’ -mantfold and 0 is an open-submantfold and 1 : 0 -+ M rhe inclusion

map then t(8) = t(M)10 and t(1) is the inclusion of f(0) in t(M).

THEOREM. The functor T defined above is a tangent bundle finctor. Moreover any two

tangent bundle,functors are natural1.v equivalent.

46. INTEGRATION OF VECTOR FIELDS

Leta:(a,b)+MbeaC’+’ -map of an open interval into a CA+‘-manifold M. We

define a Ck-map 6’ : (a, 6) -+ T(M), called the canonical lifting of 6, by a’(r) = da,(l).

We note that rcc’ = G i.e. that 6’ is in fact a lifting of 0.

DEFINITION. A CL-vector field on a Ck+ ’ -manifold M is a CL-cross section of T(M),

i.e. a CL map X: M -+ T(M) such that TC 0 X = identity. A solution curve of X is a Cl-map

u of an open interval into M such that cr’ = X 0 CT. If 0 is in the domain of the solution u w’e

call u(0) the initial condition of the solution a.

The facts stated below are straightforward consequences of the local existence and

uniqueness theorem for vector fields and proofs will be found in [4, Chapter IV].

Let M be a CL+‘-manifold (k 2 1) with dM = 4 and let X be a CL-vector field on M.

THEQREM (1). For each p E M there is a solution curve aP of X with initial condition p

such that every solution curve of X with initial condition p is a restriction of CT,.

The solution curve up in the above theorem is called the maximum solution curve of X

with initial condition p. We define t+ : M+(O,co]andt-:M+[-co,O)bytherequire-

ment that the domain of u,, is (l-(p), t’(p)). They are called respectively the positive and

negative escape time functions for X.

306 RICHARD 5. PALAIS

THEOREM (2). [‘t-(p) c s c t’(p) and 9 = a,(s) then a,, = op O T, where 5, : R 4 R

is defined by 5,(t) = s + t. In purticufur t+(9) = t’(p) - s and t-(9) = t-(p) - s.

THEOREM (3). tC is upper semi-continuous and t - is lower semi-continuous. Also if

t+(p) < co then a,(t) has no limit point in M as t -+ t’(p) and if t-(p) > -XI then a,(t)

has no limit point in M us t - t-(p).

COROLLARY. If M is compuct then tf E co and t- = -a .

To state the final and principal result we need the notion of the product of two CL-

manifolds. This is defined if at least one of the two manifolds has no boundary. If

c~ : D(p) +. V is a chart for M and tj : D(9) --+ W is a chart for N then cp x $ : D(p) x

D(l(/) + V x W is a chart in M x N (note that the product of a half-space in I’ with W

is a half-space in V x W). The set of such charts is a C’-atlas for M x N and we denote

by M x N the resulting Ck-manifold. If N has no boundary then Z(M x N) = (dM) x N.

Now we go back to our C’-vector field X on a C” ‘-manifold M with ZM = 4.

DEFINITION. Let D = D(X) = {(p, t) E M x Rlt-(p) < t < t’(p)} and for each

t E R let D, = D,(X) = {p E Ml(p, f) E D). Define cp : D -+ M by cp@, t) = a,(t) and

cPr. . D, -+ A4 by qr(p) = o,(t). The indeex-ed set q, is culled the maximum local one parameter

group generated by X.

THEOREM (4). D is open in M x R and cp : D + M is of class C’. For each t E R D, is

open in M and qr is u Ck-isomorphism of D, onto D_ t hucing cp- f us its inverse. If p E D,

andq,,(p)E D, thenp E D,+,undcp,+,(p) = CP~(CP,(P)).

$7. REGULAR AND CRITICAL POINTS OF FUNCTTONS

Let M be a C’-manifold, f: M + R a CL-function. If p E M then df, is a bounded

linear functional on Mp, If dfp # 0 then p is called a regular point off and if df, = 0 then

p is called a critical point off. If c E R thenf- ‘(c) is called a level off (more explicitly the

c-level off) and it is called a regular level off if it contains only regular points off and a

critical level off if it contains at least one critical point off. Also we call c a regular value

off iff -l(c) is regular and we call c a critical value off iff -l(c) is critical.

If f and M are C2 then there is a further dichotomy of the critical points off into

degenerate and non-degenerate critical points. We consider this next.

LEMMA. Let q be u Ck-isomorphism of an open set 0 in a Eunuch space V onto an open

set 0’ in a Bunach space V’ (k 2 2). Letf: 0’ +RbeofclussC2andletg=fOcp:O+R.

Then if dg, = 0, d’g,(o,, 0J = d’f&,(drp,(uJ, dqp(t’2)).

Proof. From the chain rule we get

dg, = df,(,, dq, and

d’gx(Ur, ~2) = d*f,&~,h), dv,(v,)) + d&,(d2cp,hv ~2)).

Putting x = p in the first equation gives dfVcpJ = 0 (because dqDp is a linear isomorphism)

and then putting x = p in the second equation gives the desired result.

MORSE THEORY ON HILBERT MA&lFOL.DS 307

PROPOSITION. I_ f is a C2-real valued function on a C’-manifold .\I and if p is a critical

point off then there is a uniquely determined continuous, sJ*mmetric. bilinear form H( f )p

on M,, called the Hessian off at p, with the folfowing propert.)‘: if9 is any chart at p

H(f),(4 fin) = d2(f 1 cp- L)vpcp,(r,, w*)

Proof. Immediate from the lemma.

Given a Banach space Y and a bounded, symmetric bilinear form 5 on V we say that

5 is non-degenerate if the linear map T: V -+ V* defined by T(L.)(H.) = B(L’, w) is a linear

isomorphism of V onto V*, otherwise B is called degenerate. Also we define the index of

B to be the supremum of the dimensions of subspaces W of V on which 5 is negative

definite. The co-index of 5 is defined to be the index of -5.

DEFINITION. If f is a Cz-real raluedfunction on a C’-manifold ,LI and p is a critical

point off we define p to be degenerate or non-degenerate accordingly as the Hessian off at p

is degenerate or non-degenerate. The index and co-index off at p are dejned respectively

as the index and co-index of the Hessian off at p..

The finite dimensional version of the following canonical form theorem is due to

Marston Morse :

MORSE LEMMA. Let f be a Cki2- real ralued function (k 2 1) defined in a conrex

neighborhood 0 of the origin in a Hilbert space H. Suppose that the origin is a non-degenerate

critical point off and thatf vanishes there. Then there is an origin preserring CL-isomorphism ~0

of a neighborhood of the origin into H such that f(cp(r:)) = ,‘PF,’ - i,(l - P)ci12 where P

is an orthogonal projection in H.

Pro05 We shall show that there is a CL-isomorphism 4 of a neighborhood of the

origin in H such that $(O) = 0 and f(c) = (A@(r), It/(r)) where (,) denotes the inner

product in Hand A is an invertible self-adjoint operator on H. The remainder of the proof

uses the operator calculus as follows. Let h be the characteristic function of [0, co). Then

h(A) = P is an orthogonal projection. Let g(l) = /A.( -I”. Since zero is not in the spectrum

of A, g is continuous and non-vanishing on the spectrum of A so g(A) = T is a non-singular

self-adjoint operator which commutes with A. Now lg(;.)’ = sgn(i.) = h(i.) - (I - h(l))

so AT2 = P - (1 - P). Then

/(II/-‘TV) = (Al’v, Tcj = (AT2c, v) = (PC, v} - ((1 - P)v, vj

= ijPc~/‘- ji(l - P)vjj2.

It remains to find $. By Taylor’s theorem with m = 2 f(r) = B(v)(~. I.) where 5 isa C’-map

of fl into bounded symmetric bilinear forms on H. Using the canonical identification of the

latter space with self-adjoint operators on H we have f(v) = (A(L.)L., r) where A is a C’-map

of B into self-adjoint operators on H. Now d2f0( I‘, hl) = 2(A(O)r. K.) and since the origin

is a non-degenerate critical point off, A(0) is invertible, SO A(L.) is invertible in a neighbor-

hood of the origin which we can assume is 0. Define B(r) = A(c)-‘A(0). Since inversion is

easily seen to be a F-map of the open set of invertible operators onto itself (it is given

locally by a convergent power series) 5 is a C’-map of 0 into L(H. H), and each B(C) is

invertible. Now B(0) = identity and since a square root function is defined in a neighbor-

C


hood of the identity operator by a convergent power series with real coefficients we can

define a C&-map C : 8 + L(I-2, H) with each C(c) invertible, if 0 is taken sufficiently small,

by C(c) = B(c)“~. Since A(0) and A(c) are self-adjoint we see easily from the definition

of B(U) that B(a)*A(c) = A(c)B(p) (both sides equaling A(0)) and clearly the same relation

then holds for any polynomial in B(c) hence for C(c) which is a limit of such polynomials.

Thus C(C)*A(L)C(~) = x4(~)C(r)~ = A(r)B(r) = A(O), or A(c) = Ci(r)*A(0)Ci(~.) where we

have put C,(c) = C(u)- ‘. If we write $(c) = C,(c)c then $ is of class CL in a neighborhood

of the origin and f(u) = (C,(r)*A(O)C,(v)~~, c) = (A(O)+(c), i(r)) so it remains only to

show that d$, maps H isomorphically, and hence, by the inverse function theorem that

$ is a C’-isomorphism on a neighborhood of the origin. An easy calculation gives

dl//, = C,(c) + d(C,),(u) so in particular d$, = C,(O) which in fact is the identity map

of H.

COROLLARY. The index off at the origin is the dimension of the range of (1 - P) and

the co-index off at the origin is the dimension of the range of P.

Proof. Let W be a subspace on which d2f, is negative definite. If w E W and

(1 - P)w = 0 then d2f,(w, w) = 2jiPw:I’ - 2j((l - P)w~:~ = 2jjPwsj;2 2 0 so w = O.Thus

(1 - P) is non-singular on W, hence dim W _< dim range (1 - P).

q.e.d.

Canonical Form Theorem for a Regular Point

Let f be a C’-real ralued function defined in a neighborhood U of the origin of a Banach

space V(k 2 1). Suppose that the origin is a regular point off and that f Eanishes there.

Then there is a non-zero linear functional I on V and an origin preserving C’-isomorphism cp

of a neighborhood of the origin in V into V such that f((p(o)) = l(u).

Proof. Let 1 = df,, # 0. Choose x E V such that l(x) = 1. Let W = {u E Vll(c) = 0).

Define T: V + W x R by T(v) = (t. - 1(0)x, l(u)). Then T is a linear isomorphism of V

onto W x R. Define $ : U + W x R by G(u) = (~1 - l(u)x, f(u)). Then I,!J is of class C’

and d$,(tl) = (u - 1(0)x, djl(a)). In particular dill0 = T so by the inverse function theorem

$-‘T is a CL-isomorphism of a neighborhood of the origin in V into V which clearly

preserves the origin. If u’ = t/r-‘TV then (D’ - l(u’)x,f(u’)) = Ic/(u’) = T(u) = (u - l(u)x,

l(u)), i.e. f($-‘TV) = l(u).

q.e.d.

DEFINITION. Let M be a Ck-manifold and let N be a closed subspace of M. We call N

a closed CL-submanifold of M if the set of charts in N which are restrictions of charts for M

form an atlas for N. This atlas is automatically C’ and we denote the CL-manifold determined

by this atlas by N also.

Smoothness Theorem for Regular Levels

Let f be a C’-real zalued function on a CL-manifold M (k 2 1). Let a E R be a

regular value of f and assume that f-‘(a) does not meet the boundary of M. Then


M, = {x E MJf(x) I a} and/--y a are closed CL-submanifolds of M and ZM, is the disjoint ) union of Ma n 2.M andf-‘(a).

Proof. An immediate consequence of the canonical form theorem for a regular point.

@. THE STROI;G TIUVSVERSALITY THEORE>I

Let M be a Ckf’-manifold without boundary (k > I), X a Ck-vector field on M and

43, the maximum local one parameter group generated by X (see 56). Iff is a Ck real valued

function on M we define a real valued function Xf on M by Xf(p) = df,(X,). In general

Xf will be of class C ‘-’ but of course in special circumstances it may be of class C’ or

C”‘. If we define h(t) = f(cp,(p)) = f(u,ct)> then h’(t) = df,,c,,(o,‘(t)) = df,l,,,(KI(,,) =

Xf(cp,(p)) so that if Xf = 1 then f((cp,(p)) = f(p) + t.

PROPOSITION. Assume that Xf E I, f(,W) = (- E. E) for some E > 0, and that q*(x)

is definedfor It + f(x)/ < E. Then W =,f- '(0) is a closed Ck-submangbld of M and the map

F : W x (- E, E) + M defined by F(w, t) = cp,(bv) is a C’-isomorphism of W x (- E, E)

onto M which for each c E (- E, E) maps W x {c} Ck-isomorphicall-v onto f -l(c).

Proof. Since at a critical point p off Xf(p) = d&(X,) = 0 the condition Xf E 1

implies that every real number is a regular value off, hence that every level f -l(c) and in

particular W is a closed C’-submanifold of iv. If F(>cl, t) = F(w’, t’) then

t =f(w) + t =f(cp,(w)) =f(cp,.(w’)) =f(rv’) + t’ = t’

hence q,(w) = cp,(w’) and since Q, is one-to-one N’ = 1~“. We have proved that F is one-

to-one. If m E M then I-f(m) + f(m)/ < E so w = cp- ,c,,,,(m) is well-defined and f(w) =

f(m) -f(m) = 0 so w E W. Moreover F(w,f(rn)) = (p,-(,,,,((~_~(,,,,(m)) = m. Hence F is

onto and moreover we see that F-‘(m) = (cp_ /(,,(m), f(m)) which by Theorem (4) of 156

is a Ck-map of M into W x (- E, E). Thus F is a Ck-isomorphism and since f(F(w, i)) =

f(cp,(w)) =f(w) + c = c the final statement of the theorem also follows.

q.e.d.

DEFINITION. A Ck-rector field X on a CkC ’ -manifold writhout boundary M (k 2 TJ will

be said to be C’-strongly transtierse to a CL-function f: M ---t R on a closed intercal [a, b]

iffor some 6 > 0 the following two conditions hold for V = f -‘(a - 6, b + S):

(1) Xflf-of class CL and non canishing on V,

(2) If p E V and cp is the maximum solution curce of X with initial condition p then

a,(t) is-defined and not in Vfor some positice t and also for some negative t.

Now given the above, V is clearly an open submanifold of M and by (1) Y = X/Xf

is a well-defined C’-vector field on V. Moreover Yf is identically one on V so the integral

curves of Y are just the integral curves of X reparametrized so that f(o(t)) = f(a(0)) + t,

hence condition (2) is equivalent to the statement that if $, is the maximum local one

parameter group generated by Y on V then Ic/,(p> is defined for a - 6 <f(p) + t -c b + 6. If we put

g = f,V-F, b-a

E=y+6


we see that the triple (V, g, Y) satisfies the hypotheses made on the triple (M,f, X) in the

above proposition. This proves

Strong Tmnsversality Theorem

Let f be a Ck real valued function on a Ckc ’ -manifold without boundary M (k 2 1).

If there exists a Ck-vector field X on M which is CL-strongly transverse to f on a closed interval

[a, b] then W = f- ‘( a is a closed Ck-submanifold of M and for some 6 > 0 there is a CL- )

isomorphism F of W x (a - 6, b + 6) onto an open submamfold of M such that F maps

W x {c} C’-isomorphically onto f-‘(c) for all c E (a - 6, b + 6). In particularf-‘([a, b])

is CL-isomorphic to W x [a, b].

COROLLARY. There is a Ck-map H : A4 x I 4 M such that if we put H,(p) = H(p, s)

then:

(1) H, is a CL-isomorphism of M onto itselffor all s E I,

(2) H,(in) = m ifm $f -‘(a - 6/2, b + 6/2);

(3) H, = identity;

(4) H,(f-‘(-~,a]) =f-‘(-co, b].

Proof. Let h : R + R be a C”-function with strictly positive first derivative such that

h(t) =. t if t # (a - 6/2, b + 6/2) and h(a) = 6. Define H, = identity in the complement of

f -‘(a - 6/2, b + 6/2)anddefine H,inf-‘(a - 6, b + 6) by H,(F(w, t)) = F(w, (1 - s)t +

sh(r)).

$9. HLLBERT AND RIEMANMAN MANIFOLDS

Let A4 be a CL+‘-Hilbert manifold (k 2 0), i.e. M is a C’+‘-manifold and for each

p E M M, is a separable Hilbert space. For each p E M let (,)P be an admissible inner

product in Mp, i.e. a positive definite, symmetric, bilinear form on M, such that thenorm

llullp = (u, v)“’ defines the topology of M,. Let rp be a chart in M having as target a

Hilbert space H with inner product (,). We define a map G’ of D(q) into the space of

positive definite symmetric operators on H as follows: if x E D(q) then dq-’ is an iso-

morphism of H onto M,, hence there is a uniquely determined positive operator GQ (x)

on H such that (G’ (x)u, v) = (dq; t(u), dqo; r(v)),. Suppose II/ is another chart in M

withtargetH.-Let U=D(cp)nD(~)andletf=cpoJ/-‘:~(U)-rcp(U),sod~;’=

dqo;’ 5 d&z, for x E U. Then

<G’(x)u, u> = <dK’ d&&), dK ’ d&W,

= <G’(x) dfs&)v df,J4),

hence G’(x) = df ft,,G’(x) df+(=,, x E U. Since f is of class C’+’ it follows that if G’ is of

class CL in I/ then so also is G’. Hence it is consistent to demand for each chart Q, that GQ

is of class CL. If this is so we will call x --, (,), a CL Riemannian structure for M, and M

equipped with this extra structure will be called a Ckfl-Riemannian manifold. We will


maintain the notation used above. That is given a chart cp in a Ck-Riemannian manifold

M we will denote by G” the function defined above, and in addition we define R+’ on

D(p) by P(x) = (G’(x))-‘. Also we define a function *I / in T(M) by “c;; = (r, r)‘;’

for u E M,,. Clearly ‘i 1,’ is of class CL on T(M), hence ! :, is continuous on T(M) and of

class C’ on the complement of the zero section. If cr : [a, h] -+ M is a Cl-map then

t + iiu’(t);/ is continuous on [a, b] hence

L(a) = ‘,,o’(t)!, dr J 0 is well defined and is called the length of 0. It is easily seen that if x and )’ are two points

in the same component of M then there exists a C’-curve joining them, hence we can define

a metric p in each component of M by defining p(x, y) to be the infimum of the lengths of

all Cl-paths joining x and y. It is clear that p is symmetric, satisfies the triangle inequality,

and is non-negative. That p(x, ~1) > 0 if x # )’ and is hence a metric, and that the topology

given by this metric is the given topology of M follows easily from the following two lemmas:

LEMMA (1). Let H be a Hilbert space, f: [a, b] - H a C’-map. Then

J b

llf’(f)ii dt 2 iIf(b) -f(a)il. *

Proof. We can suppose f(b) # f(a). Let g(t)(f(b) - f(a)) be the orthogonal projection

off(t) -f(a) on the one-dimensional space spanned by f(h) -f(a). Then g : [a. h] -+ R

is C’, g(a) = 0, g(b) = 1 and f(t) -f(a) = g(t)(f(b) -f(a)) + h(t) where h : [a, h] -+

(f(b) -f(a))’ is C’. Then f’(t) = g’(tKf(b) - J(a)) + h’(t) where h’(t) 1 (f(b) - f(a)).

hence

so

llf’(t)l~* = :,.f(b) -f(a~i~*~ld(Ol* + W(t)ll* L ll,f(hl -

But

Ilf’(~)il dt 2 Ill(b) -./‘(ahi . J b

k ([Ii dt. a

q.e.d.

LEMMA (2). Let H be a Hilbert space, p E H, and G a continuous map of a neighborhood

of p into the space of posirive operators on H. Then there exists r > 0 such rhat G is d@ined

on B,(p) and positire constants K and L such that.

(1) yf: [a, b] + B,(p) is a Cl-map withf(a) = p


(2) iff: [a, b] + His a C’-map withf(a) = p and c = Sup {t E [a, b]]f([a, r]) c B,(p)}

then

s c(c(f(t))f’(t),j’(t))lIZ dt > Lr. a

Proof Let Q(x) = G(x)-‘. By continuity of G and Q we can choose K and L > 0

such that liG(x)ll < K2 and liQ(x)ll < Le2 for x in some open ball B,@). Then

(G(f(r))f’(r)9f’(r)> S K’U’(r)>f’(r)> and

(f’(r)J’(r)> = Wf(r))G(f(r))f’(r)J’(r))

I L- 2<G(f(r))f’(r)7 f’(r)>

if f(r) E B,(p). Then (1) follows immediately while (2) follows from (1) and Lemma (1). q.e.d.

DEFINITION. rf M is a Ck+’ -Riemannian manifold then the metric p defined above on

each component of M is called the Riemannian metric of M. If each component of M is a

complete metric space in this metric then M is called a complete C” ‘-Riemannian manifold.

If’b is a Cl-map of an open interval (a, b) into a Riemannian manifold M we define the

length of Q, L(a),,to be

lim ‘/a’(r)l\ dr. i (1-o ;I

B-b

Of course L(a) may be infinite. However suppose L(a) < co. Given E > 0 choose t, = a < t, < . . . < t, c b = tn+l so that

ft+ I [la’(t)jl dt < E.

li

Then clearly a((a, 6)) is included in the union of the s-balls about the a(t,) i = 1, 2, . . . . n.

Thus

PROPOSITION (1). Zf M is a C’+ ’ -Riemannian manifold and o : (a, b) + M is a Cl-curve

of finite length, then the range of CT is a totally bounded subset of M, hence has compact

closure if M is complete.

PRORXITION (2). Let X be a Ck-vector field on a complete Ckf ‘-Riemannian manifold

M(k 1 1) and let cr : (a, b) -+ M be a maximum solution curve of X. If b c co then

I

b

II-WWI dt = 00 0

(in particular [/X(17(t)) I( is unbounded on [O, b)) and similarly if a > - co then

I ollX(o(t))/l dt = 00

(in particular [lX(u(t)) // is unbounded on (a, 01).

MORSE THEORY ON HILBERT MAhIFOLDS 313

Proof. Since a’(t) = X(a(t)) it follows from Proposition (1) that if

s

b

llX(d4>ll dl < ~0 0

then a(t) would have a limit point as t --+ b, contradicting Theorem (3) of $6. q.e.d.

Now letf: M -+ R be a CL+’ real valued function on a Ckf ‘-Riemannian manifold M.

Given p EM df, is a continuous linear functional on )%I,, hence there is a unique vector

Vf, E M, such that dfp(r) = (L., Vf,), for all L‘ E iv,. Vfp is called the gradient off at p and Vf: p --) Vf, is called the gradient off. We claim that Vf is a CL-vector field in M.

To prove this we compute it explicitly with respect to a chart p : D(q) + H where H is a

Hilbert space with inner product, (,). Let T be the canonical identification of H* with

H, so if I E H* then I(c) = (L., Tl). Since T is a linear isomorphism it is C”. Define g on

the range U of c~ by g = f 3 c+o-l. Then g is of class Ckr ’ hence dg : Cl -* H* is class C’

so To dg = i. is Ck. Now by definition of C”

(G’(x) dq,(Vf,), u> = Vf,, dq; ‘(u)>, = dfx,dv; ‘(~1

= dg,c,,(v) = <T dq,,,,.o)

so dp,(Vf,) = Q@‘(x),J(cp(x)). Since x -+ G’+‘(X) and hence .r ---, G’+‘(x)-’ = Q’+‘(x) are Ck

it follows that x + dp,(Vf=) is a Ck-map of O(q) into H. By definition of the C’-structure

on T(M) this means that Vf is a CL-vector field on A4. We note the following obvious

properties of VA First Vf, is zero if and only ifp is a critical point off, so the critical locus

off is just the set of zeros of the real valued function i]Vf ;i. Moreover

(V_f>f(~> = df,Plf,) =

so (Vf)fis positive off the critical locus off.

410. TWO-THIRDS OF

In this section we assume that A4 is a

THE MAINTHEOREM _^

C ‘+‘-Riemannian manifold (k > 1) without

boundary and that f: M --r R is a Ckc2 -function on M having only non-degenerate critical

points and satisfying condition

(C) If S is any subset of M on which f is bounded but on which $Vf !j is not bounded

away from zero then there is a critical point offadherent to S.

We note that it is an immediate consequence of the Morse Lemma of $7 that a non-

degenerate critical point of a C3 function on a Hilbert manifold is isolated. In particular

the critical locus off is isolated. We will now prove that much more is true. Let a < b

be two real numbers and suppose that {p,} was a sequence of distinct critical points off

satisfying D <f@,) < 6. Choose for each n a regular point qn such that

P(4”? A> < $ IIVf4nI/ < i and * <f(q.) < b.

Then by condition (C) a subsequence of the {qn} will converge to a critical point p off:


But clearly the corresponding subsequence of {pm} will also converge to p, contradicting

the fact that critical points offare isolated. Hence

hOPOSITION (1). If a and b are two real numbers then there ure uf most a finite number

of critical points p off satisfying a < f(p) < b. Hence the critical ralues off ore isolated and

there are ut most a finite number of critical points off on an! critical level.

LEMMA. Assume M is complete and let u : (a, p) d M be a maximum solution curre of

VJ Then either lim f(o(t)) = co or else /3 = 00 and a(t) has u critical point off us a limit

point as t + /I. %zilar/y either lim f(a(t)) = - KJ or else I = -co und a(t) has u critical

point off us a limit point us t 4 :. ’

Proof. Let g(t) = f(a(t)). Then

s’(l) = %,,,(a’(r)) = %(,,(VJ-e(t)) = IlVfa,r,l~~~

Thus g is monotone, hence has a limit B as t 4 /I. Suppose B c co. Then since

g(1) = g(O) + s

‘g<(s) ds = y(O) + ‘liO/,r.lli2 ds

0 I 0

it follows that

s

B liWd2 ds < ~0.

0

By Schwartz’s inequality we have

so fl < co would contradict Proposition (2) of 99. Hence j = x. But then clearly

liVf~,sjij cannot be bounded away from zero for 0 2 s < co since then the above integral

could not converge. Since f(a(s)) is bounded for 0 5 s < co (and in fact lies in the interval

[f(a(O)), B] it follows from condition (C) that a(t) has a critical point off as limit_point

as t + j?.

PROWSITION (2). If M is complete and f has no critical values

[a, b] then Vf is C ‘+I-strongly transverse to f on [a, b] hence by the

Theorem (§8) M. = {x E MI f(x) I a} and Mb = {x E MI f(x) 5 b}

Proof. By Proposition (1) of this section there is a 6 > 0 such

in the closed intercal

Strong Trunsuersality

are CL+’ -isomorphic.

that f has no critical

values in [a - 6, b + 61. Let V = f-‘(a - 6, b + 6). Then (Vf )f = /iVf ii2 is strictly

positive and C’ ’ ’ in V. Let p E M and let 0 : (a, /I) 4 M be the maximal integral curve of

Vf with initial condition p. We must show for some z < t, -z 0 < t, < /I that a(tl) and

a(tJ are not in V, i.e. that f(a(tl)) 5 a - 6 and f(a(t,)) 2 b + 6. Suppose for example

that f(a(t)) < b + 6 for 0 < t < /I. Then by the lemma a(t) would have a critical point p

as limit point as t + j3. Since f is continuous and f(o(t)) monotone it follows that

a - 6 <f(O) I f(p) I b + b so f(p) would be a critical value off in [a - 6. b + 61, a

contradiction.

q.e.d.


Before completing the Main Theorem we must discuss the process of adding a handle

to a Hilbert manifold.

$11. HAiimXs

Let LY denote the closed unit ball in a separable Hilbert space H of dimension k

(0 2 k I co). Note that sincef: H + R defined byf(x) = 1ix’;’ is a C” real valued function

in H and zero is the only critical value ofj; it follows from the Smoothness Theorem for

Regular Levels (57) that P = f-‘( - co, l] is a closed C” submanifold of H. Moreover

the boundary ZD’ of Dk is Sk- ‘, the unit sphere in H. We call Dk x D’ a handle of index k

and co-index I. Note that ti x D’ is not a differentiable manifold since both p and D’

have non-empty boundaries (unless k or f = 0). However if we put bk = Dk - dDk

then both Sk-’ x D’ and dk x D’ are Cm-Hilbert manifolds.

DEFINITION. Lef M be a C’-Hilbert manifold and N a closed submanifold of M. Let.f

be a homeomorphism of Dk x D’ onto a closed subset h of M. We shall write M = N y h

and say that M arises from N by a C’-attachment f of a handle of type (k, 1) I$

(1) M= Nub.

(2) f IS’-’ x D’ is a C’-isomorphism onto h n d/V,

(3) f Ibk x D’ is a C’-isomorphism onto M - N.

Suppose we have a sequence of C’-manifolds N = N,, N,, . . . . N, = M such that

Ni+ I arises from Ni by a C’-attachment fi of a handle of type (ki, li). If the images of the

fi are disjoint then we shall say that M arises from N by disjoint C’-attachments (f,, . . ..j.)

of handles of type ((k,, I,), . . . . (k,, /,)).

With the next lemma and theorem we come to one of the crucial steps in seeing what

happens when we “pass a non-degenerate critical point”.

LEMMA. Let i. : R -+ R be a C” function which is monotone non-increasing and satisfies

A(X) = 1 if x I l/2, J.(x) > 0 g x < 1 and l.(x) = 0 if x 2 1. For 0 I s I 1 let &) be

the unique solution of A(a)/1 + u = +(I - s) in the interrlal [0, I]. Then o is strict/y

monotone increasing, continuous, C” in [0, 1) and a(O) = l/2, a(l) = 1. Moreover if E > 0

and I? - v2 2 -E and u2 - t’2 - (3c/2)A(u2/e) _< - E then

V2 U21&U 2 ( j E+U .

ProoJ Clearly A(a)/1 + u is strictly monotonically decreasing if 0 5 c 5 1. Since

it is one for u = 0 and zero for u = 1 the theorem that a continuous monotone map of an

interval into R has a continuous monotone inverse gives easily that u exists, is continuous

and monotone. That a(O) = l/2 and u(l) = 1 is clear and since A(u)/1 + d has a non-

vanishing derivative in [0, 1) it follows from the inverse function theorem that cr is C” in

[0, 1). Now consider f(u, u) = u2 - .su(u~/(E + 2)) in the region

. u2 -v21 -&, u2 - v2 _;I” 5-6. ( 1 E


For L; fixed f is clearly only critical for u = 0 where it has a minimum, hence f must assume

its maximum on the boundary. On the boundary curve u2 - o2 = -E we have

t.2/(& + II’) = 1 so f(u, L’) = u2 - E. If (u, o) is not also on the other boundary curve then

so u2 < E so f(u, r) < 0. On the other hand

we have

u2 - v2 -;

v2 .

Now on this boundary

and clearly

By definition of a(p)

if (u, V) is on the boundary

d(U 2;&) = -E

3

z = ’ - 2(1 + U2/E) * EfU

U2/& 2 l/2 otherwise v2 <o Ef

2 U2 - I1 so - = a(pj.

E E

V2 z = l _ 2 44P)) .s+u

-=1-((I-p)=p 2 1 + 4P)

hence

f(u, v) = u2 - EU & ( 1 = dP> - &a>,

i.e. f vanishes on this boundary. Thus f I 0 everywhere on the boundary of the region

and hence also in the interior. -_ q.e.d.

THEOREM. Let B be the baN of radius 2 E about the origin in a Hilbert space H. Define

f: B -+ R by f(v) = llPvlj2 - jlQv~~’ w h ere P is an orthogonal projection on a subspace H’

of dimension I and Q = (1 - P) is a projection on a subspace Hk of dimension k. Let

g(v) =f(v) - 4 411P412/4

where I : R --* R is as in the lemma. Then M = {x E Bjg(x) I - E} arises from N =

{x E BI f(x) 5 - E} by a C” attachment F of a handle h of type (k, I).

Proof. Before commencing on the proof we give a diagram of the case k = I = 1 (Fig. 1).

Let ok and D’ be the unit discs in Hk and H’ respectively. Let h be the set in B where

f2 -eandg< - .ssoM=NuhandNnhc8N. DefineF:D’x D’-+Hby

F(x, y) = (Ea(llxli2)llyii2 + 8)lj2x +(m(ilx112>j1’2y.


f =-tz

____- H[ --

FIG. 1

Where d is as in:the lemma. Then,

_f(Ux, y>> = ~[4d12~llA12 - (1 + ~~ll~li2~IIYl12)ll~l12]

= 4~(llxl12>llYl12(~ - Ilxl12> - 114121 z --E

g(F(s,Y)) = 4~(ilxl12>ilYl12(~ - llxl12> - 11~~112

- 3~.(~~ll~I12~llYl12~]

Since 1 is monotone decreasing

g(F(x, Y)) 5 &[fdilxl12Xl - llxl12~ - lIxi12 - 344ixl12>)] __

but E.(e(lix[12)) = +(l + o(ilxl12))(l - llx/12) by definition of o; substituting we see that

g(F(x, y)) I - E, hence F maps D’ x D’ into h. Conversely suppose w E h and let u = Pw,

v = Qw so jju!12 - llujj2 2 -E and

3E [/njj’ - IIui12 - - ~(I~u]~~/E) 5 --E.

2

Then ~~zI/~~/(E f IIuI/~) 5 1 so x = (E + j/~11~)-“~u E Dk. Also a(llul12/(s + lluii2)) is well

defined and by the lemma l/uii2/ea(llvl12/(s + ilu~~‘)) 5 1 soy = (~a(ilvl/‘/(~ + Il~lI/~)))-~‘~tl

E D’. Thus

G(w) = ((E + ilPwl12)-“2Qw, (E~(IIQwII~/(E + IIPwli2)))-1/2Pw)

defines a map of h into Dk x D’. It is an easy check that F and G are mutually inverse

maps, hence F is a homeomorphism of Dk x D’ onto h. From the fact that d is C” with

non-vanishing derivative in [0, 1) it follows that F is a Cm-isomorphism on B’ x D’. On

Sk-’ x D’ F reduces to F(x, y) = (E(/IyI(2 + 1))“2X + &1’2y


which is clearly a C”-isomorphism onto N n h, the set where ,f = - E and !PH~!;’ I E.

q.e.d.

$12. PASSLNG A

section we will complete the proof of the main by analyzing

happens as we pass a critical level. We will need:

LEMMA. Let Hilbert

space H, Q positire definite. product (,) in H such

that Q(D, c) = (Gc, r) and f(o, c) = - li(l - P)ril’ where P is an orthogonal

jection which commutes with the positine

proving the Morse lemma. Note

that Q(u, L’) is an admissible product in H, hence f(r, 1%) = Q(AP, r) where A is an

invertible adjoint with to this inner product. Let G = (Al-’ and

P = h(A), where h is the characteristic Q(lAlu, r)

so Q(u, t.) = (Gu, v). It is clear that any function of A is self relative to (,) so P

is an orthogonal projection and G a positive

IlPall’ - I:(1 - P)ql’.

q.e.d.

We now return to the situation

coindex off at pi. By the Morse

Lemma (97) we can find for some d < 1 a CL-chart cp i at pi whose image is the ball of radius

26 in a Hilbert space H, such that qr(pi) = 0 and fqf’(u) = llPivli2 - il(l - Pi)Vij2

where P, is an orthogonal projection in Hi of rank I, and (1 - Pi) has rank ki. Moreover

by the above lemma if G’ is the positive operator in Hi defined by (d(p,‘(u), d(p,‘(u)),, =

(G’u, v) then we can assume that G’ commutes with Pi. This will be crucial at a later point

in the argument.

By Proposition (1) of $10 we can choose E < a2 so small that 0 is the only critical value

off in (- 3e, 3~). Let W = f -‘( - 2.5, co). We define a CL-real valued function g in W by

3, . g(cPt’ ‘iv)) =f(V; ‘tv)) - 2 A(llpivl12iE)~

where L is as in the lemma of $11, and g(w) = f(w) if w 4 ; D(cp,). Note that if w = ‘P;](u) I=1

E Wandf(w) 9 g(w) then 1(IIPit~li2/s) + 0 so \~P,zJ/~~ < E (hence f(w) < E) and IIP,ul12 -

ll(I - Pi)Ul12 = f(w) > - 2s SO 111~11~ = \IPivl12 + ll(I - Pi)~I12 < 4~ c 4a2, so the

closure of {w E Wn D(cp,) If(w) * g(w)} . IS included in the interior of D((pJ, which proves

that g is C”. The above also shows that {w 4 W If(w) 5 E} = {w E W I g(w) I ~1.

Now it follows immediately from the theorem of $11 that {w E W/g(w) 5 - E} arises from

{WE Wf(w) I - E} by the disjoint CL-attachment of r handles of type (k,, It) . . . .

(k,, I,). We will prove:


LEMMA. If .5 is suficiently small then Vf is CL-strongl_v transrerse to g on [ - E, E].

Itthenfollowsfrom the strong transversality theorem that there exists a C’-isomorphism

h of W onto itself such that h(w) = w if lg(w)] L -3~/2 and h maps {w E W/g(w) I -E}

C”-isomorphically onto {w E W/g(w) I E} = {w E W/f(w) _< E}. We can extend h to a

C“-isomorphism of M by defining k(x) = x if x $ W. It follows that {x E Ml f(x) 5 E)

is C’-isomorphic to {x E MI f(x) < - E} with r-handles of type (k,, /,), . . . . (k,, I,) disjointly

CL-attached. More generally by applying Proposition (2) of $10 to the intervals [LZ. - E]

and [E, b] we get the third part of the main theorem.

THEOREM. Let f be a Ck+’ real raluedfunction on a complete Ck+ 2-Riemannian manifold

U(k 2 1). Assume that al/ the critical points off are non-degenerate and that f satisfies

condition (C). Let pl, . . . . p, be the distinct critical points off on f -l(c) and let ki and Ii be

the index and coindex off at pi. If a < c < b and c is the only critical value off in [a, b]

then (x E MI f(x) I b} is C”-isomorphic to {x E MI f(x) _< ai with r-handles of type

(k,, II), . . ., (k,, I,) disjointly CL-attached.

It remains to prove that Vf is CL-strongly transverse to g on [- E, E]. if E is sufficiently

small. Let

V = [XE Wi - 5E < g(x) < 2). 4 4

We note that since

and f has no critical value in (-_j~, 3.5) except zero, the only possible critical points off

in vcould bep,, . . . . p,. But

so f has no critical points in i? NOW let p E V and let r~ : (a, /I) + M be the maximal integral

curve of Vf with initial conditionp. Then by the lemma of $10 eitherf(a(t)) -+ co as t -+ /I,

so a(t) gets outside Y as t + B or else a(t) has a critical point off as limit point as t -+ p

so again a(t) must get outside V as t -+ p. Similarly a(t) must get outside V as t -+ CL.

Thus it remains only to show that (Vf)g is CL and positive in V. Outside v D(cp,), f = g i=l

so (Vf )g = (Vf )f = iiVf I]* which is CL+’ and is positive since f has no critical points in Y.

What is left then is to show that (Vf)g is Ck and does not vanish on D(qi) except at pi.

The following proposition settles this local question.

PROPOSITION (1). Let 0 be a neighborhood of zero in a Hilbert space H with inner product

( ), mude into a C’+’ -Riemannian manifold (k 2 0) by defining (u, o), = (G(w)u, O>

where G is a Ck- map of 8 into the incertible positive operators on H. Let P be an orthogonal

yrojection in H which commutes with G(0) and define f(u) = j/Pr,/j* - ]I(1 - P)rl!l* and

g(v) = f’(u) - ?J A( I; Pull */&)


where I is as in the lemma of $11. Then (Vf )g is Cc and for E suficiently small does not

canish on the 2 E ball about the origin except at the origin.

Proof. Let n(x) = G(x)-’ so that Q(0) also commutes with P. Let T(x) = PR(x) - R(x)P. Note that IIPxll I jjx/j and /1(2P - l)xll = I!x;/ so

(Px, R(x)(2P - 1)x) = (Px, PR(x)(2P - 1)x)

= <Rx, T(x)(2P - 1)x) + (Ps, R(x)Ps)

2 (Px, T(x)(2P - 1)x)

2 - II T(x)/1 . II4 2.

Now IIull* = (u, G(x)!A(x)u) I \iG(x)!j < u, !A(x)u) hence

((2P - Z)X, R(x)(~P - 1)s) 1 jiG(,x)il-‘~I/~\i~.

Since /IT(O)// = 0 while iIG(O)II-’ > 0 we can find a neighborhood U of the origin?

independent of E, such that for x E U

IIGWli - ’ > 3ii Vx)ll supli’l.

Since A I 0 it follows that for x in U

4(((2P - Z)x, R(x)(2P - Z)x) - jA’( l)Pxj12/e) (Px, R(x)(ZP - 1)x))

2 3(//G(x)li-’ - 311’(11Pxii*/~)I~IITx(i)lixlj*.

which is positive unless x = 0. Since the left-hand side is clearly CL it will suffice to prove

that it equals (Vf)g. From the definition of f and g df,()t) = 2((2P - Z)x, y) =

2(R(x)(2P - Z)x, y), so Vf, = 2R(x)(2P - Z)x while

dg,(y) = df,(y) - 3~‘(lIPxllz14~Px~ Y>

= 2(((2P - 1)x, y} - *lL’(ilPxI12/E)(Px, y)).

Since VfJg) = dg,(VfJ the desired expression for (Vf)g is immediate. q>e.d.

This completes the proof of the Theorem. We now consider an interesting corollary

of the proof of Proposition (1). Maintaining the notation of the proof let us define

p(x) = ll~llZ = llPxl12 + ll(l - P)xjl’ so that (f - p)(x) = -2/j( 1 - P)xll’ and

(VIXf - p)(x) = 8((P - 0x, Q(x)(2P - Z)x>

= 8((P - I)x, R(x)(P - 1)x) + 8((P - 1)x, R(x)Px).

Since R(x)Px = -PPR(x)x - T(x)x and since PQ(x)x is orthogonal to (P - Z)x we get

(Vf)(f- p)(x) = 8((P - Z)x, R(x)(P - Z)x) - ((P - Z)x, T(x)x).

Recalling the inequality (u, Q(x)u) 2 /jG(x)/j-’ . ~Iu/\~

(V/U- P)(X) 2 8iKp - Oxll(ll(P - Ox-ll.liG(x)li-’ - i/~ll~il~(x~ii). Since IIZ’(O)I\ = 0, in a sufficiently small neighborhood of the origin we have /IT(x)!: 5

f IiG(x)lj-’ so in this neighborhood

(Vf>(.f- p)(x) 2 8(I(P - I)x~I.IIG(~)I~-~(I(P - I)x;I -F).


Iff(x) 5 0 then “(P - I)x~I’ 2 ‘il’xii’ SO 2ii(P - r)xii2 2 :/xii2 which implies that

ii(p - {)X/j 2 !!Z!! 2’

hence near the originf(x) I 0 implies (V’)(f - p)(x) > 0. It follows thatf - p is mono-

tonically increasing along any solution curve of V’which is close enough to the origin and

on whichfis negative. Since clearlyf(x) > -p(x) we see that if E is sufficiently small and

a(r) is the maximum solution curve of Vf with initial condition p, where p(p) < 42, then

p(a(t)) > E impliesf(a(t)) > 0. This proves

PROPOSITION (2). Let f be a C’-real valued function on a C3-Riemannian manifold M

and let p be a non-degenerate critical point off. Then if iJ is any neighborhood of p there is a

neighborhood 0 of p such that if a is a maximum solution curce of Vf haring initial condition

in 0 then for t > 0 either a(t) E U orf(a(t)) > f(p).

COROLLARY. If a is a maximum solution curve of Vf and ifp is a limit point of a(t) as

t --, co (t -+ -co) then lim a(t) = p ( lim a(t) = p). 1-13 I-.--P

PROPOSITION (3). Let M be a complete C3-Riemannian, f a C3-real l,aluedfunction on M

which is bounded aboce (below), has only non-degenerate criticalpoints, and satisfies condition

(C). If a is any maximum solution curce of Vf then lim r~(t)( lim a(t)) exists and is a critical z-3) I---?)

point off

Proof. An immediate consequence of the above corollary and the lemma to Proposition

(2) of $10.

513. THE MANIFOLDS ff~(I, V) AND fi( V;P, Q)

In this section we will develop some of the concepts that are involved in applying the

results of the preceding sections to Calculus of Variations problems.

A map CJ of the unit interval I into R” is called absolutely continuous if either andhence

both of the following two conditions are satisfied:

(1) Given E > 0 there exists 6 > 0 such that if

then

0 5 t, < . . . < t2t+l s 1 and iiOlt2i+ 1 - t2ii < 6

(2) There is a g E L’(I, R”)

i.e. g is a measurable function from I into R” and ~iig(i)lI dt < a)

such that

a(t) = a(O) -I- s

‘g(s) ds. 0


The equivalence of these two conditions is a classical theorem of Lebesque. From the

second condition it follows that a’(t) exists for almost all t E 1, that CT’ (=g) is summabie and

a(t) = a(O) + I

‘o’(s) ds. 0

From the first condition it follows that if p is a Cl-map of R” into R”, or more generally

if cp : R” --) R” satisfies a Lipshitz condition on every compact set, then cp 0 G is absolutely

continuous. .

For reasons ofconsistency that will become clear later we will denote the set of measure-

able functions cr of I into R” such that

i

1

Iia(r)l12 dt < cc by HotI, R”), 0

rather than the more customary L2(Z, R”). Then Ho(& R”) is a Hilbert space under pointwise

operations and the inner product (,). defined by

ia, ~,>o = I

‘<otr). ~(4) dr 0

where of course (,) is the standard inner product in R”.

We will denote by H,(I, R”) the set of absolutely continuous maps o : I -_, R” such

that 0’ E H,(I, R”). Then H,(I, R”) is a Hilbert space under the inner product (,)r defined

by (a, p)t = (a(O), p(O)) + (cJ’, P’>~ and in fact the map R” @ H,(\, R”) -+ H,(f, R”)

defined by (p, g) --) 6, where

is an isometry onto.

a(t) = p + s

‘g(s) ds, 0

DEFINITION. We define L : H,(f, R”) + H,(I, R”) by La = 0’ and we define

H:(I, R”) = {a E H,(I, R”)la(O) = a( 1) = O}. -_

Then the following is immediate:

THEOREM (1). L is a bounded linear transformation of norm one. H:(I, R”) is a closed

linear subspace of codimension 2n in H,(I, R”) and L maps H:(I, R”) isometrically onto the

set of g E H,(I, R”) such that

1

1 g(r) dt = 0,

0

i.e. onto the orthogonal complement in H,(I, R”) of the set of constant maps of I into R”.

THEOREM (2). If p E H:(Z, R”) and /I is an absolutely continuous map of I into R” then

J 1

(i'(r), p(l)) dt = (;C, -Lpj,. 0

Proof. Clearly t + (A.(t), p(t)) is an absolutely continuous real valued function with

derivative (A’(t), p(t)) + (I(r), p’(t)). Since an absolutely continuous function is the

MORSE THEORY ON HILBERT MA?;IFOLDS 323

integral of its derivative and since (i(r), p(t)) vanishes at zero and one, the theorem follows.

q.e.d.

We shall denote the set of continuous maps of I into R” by C’(l, R”), considered as a

Banach space with norm ,I 1/B defined by i:u,j, = sup{ ;,a(r) It E Z]. We recall that by the

Ascoli-Arzela theorem a subset S of C’(l, R”) is totally bounded if and only if it is bounded

and equicontinuous (the latter means given E > 0 there exists 6 > 0 such that if 1s - I] < 6

then /g(s) - g(t)] < E for all g E S). Since the inclusion of C’(I, R”) in H,(/, R”) is clearly

uniformly continuous it follows that such an S is also totally bounded in H,(I, R”).

and Schwartz’s inequality completes the proof.

The following is a rather trivial special case of the Sobolev inequalities:

THEOREM (3). Ifo E H,(I, R”) then

II40 - o(s)/ 5 It - ~l”211~410.

Proof. Let h be the characteristic function of [s, r]. Then

/ia - g(s)j] = 11 IIo’(x) drl/ I [,fIlc’(.x)i] dx

= I

?~(x);]g’(x)l, dx 0

q.e.d.

COROLLARY (1). IfcrE H,(I, R”) fhen I;G\/~ 5 2ijaj1,.

Proof. By definition of /I I/, we have /ID(O 5 !/c!i, and $Lal/, I IICT/!,. Now

jla(t)!i 5 iIu(O)II f lb(t) - a(O)!1 and by the theorem ila(t) - a(O 5 ilLail,.

q.e.d.

COROLLARY (2). The inclusion maps of H,(f, R”) into C”(I, R”) and H,(I, R”) are

completely continuous. -_

Proof. Let S be a bounded set in H,(I, R”). Then by Corollary (1) S is bounded in

C”(f, R”) and by the theorem S satisfies a uniform Holder condition of order l/2, hence is

equicontinuous. q.e.d.

THEOREM (4). If cp : R” + RP is a CL’ 2-map then u --* cp D Q is a CL-map (p : H,(I, R”) -+

H,(I, RP). Moreocer $1 < m -< k then

d”&(ll,, . . , k,,(t> = d”cp,&.,(t), , . . , i,,,(r)).

Proof. This is a consequence of the following lemma ifwe take F = d’q 0 I s _< k - 1.

[Note that in the lemma ifs = 0 then we interpret L”(R”, RP) to be Rp.]

LEMMA. Let F be a Cl-map of R” into L*(R”, RP). Then the map F of H,(I, R”) into

L”(H,(I, R”), H,(I, Rp)) defined by

F(4(&, . . . , .u(G = F(49)(Mt), . . . , A(9)

is continuous. Moreover ifF is C’ then F is C’ and dF = fl.

D


Proof. We note that

(F(a)(i,, . . . , 4))‘(f) = dF,,,,(a’(t))(il(t), . . . , j.,(t)) + s ~(a(t))(;.~(t), . . . , j.:(f)), . . . , L,(I)) I= 1

hence

IV(c)G,, . *. 9 Q)‘(t)ll I lIdF,,,,ll * IIo’(t)ll. lii.,(t)il . . . lii.,(t)il

+ i~lllr(~~~~)ll~ iljLl(t)!l . . . ilj.l(t)ll ,.. IIi.,(t)~l.

Since IIli’i, I 2’;iiiir (Corollary (1) of Theorem (3)) we see that i:(F(a)(i.,, .._. j.,))’ 0

< 2’L(a)liJ.,\l, . . . IlJ.Jt where L(o) = SuplidF,(,, II. j;e’,iO + s Sup,;F(a(t))~I. Also ‘F(a)

(i 1, . . . . A,)](, I 2’ S~pIlF(a(t)):i- i!I!,Il, . . . jjik,ll,. Recalling that !‘p)l: = lip(O) i2 -I- ‘p’ ,6

we see iiP(a)(j.,, . . . . ,7.,)i(1 5 K(~7i~A,l/, . . . $QII. Since F;(a) is clearly multilinear it

follows that F;(a) E LS(H,(I, R”), H,(I, RP)). If p E H,(I, R”) then

ll(F(4 - F(P))(i, 1 . . . , &Iil m r: 2” SupilF(40) - Wf))ll . II j., II 1 Ild,il ,

and a straightforward calculation gives

llwd - F(P)@,7 . *. , J.,J)‘ll0 5 2”bf(a, p)llj.Ill, . . . IIQ1 where

M(a, P) = Swlld~,~,~lI~ ilcf - ~‘11~ + SupildFac,, - d~,~,~lI . lI~‘ll~ + 5 SupllF(a(Q) - Fb(t))li so

IIIRd - %)llI 5 K(a, P)

where III III is the norm in L”(H,(T, R”), H,(I, RP)), and K(a,p) -+ 0 if SupllF(a(r)) -

Ml)) IL Sup IldFr,c,, - dF,t,,I! and 11~ - p’ll,, all approach zero. But if p -) 0 in

H,(I, R”) then 110’ - p’I10 I !/G - pjjl goes to zero and by Corollary (1) of Theorem (3)

p + a uniformly, hence since F and dF are continuous F(p(r)) + Fo((t)) uniformly and

dF,,,, --, dF,,,, uniformly, so K(a, p) -+ 0. Thus j/IF(o) - F(p)III + 0 so F is continuous.

This proves the first part of the lemma. Now suppose F is C3 so dF is C*. By the mean

value theorem there is a Cl-map R : R” 4 L’(R”, L”(R”, RP)) such that if x = p + cthen

F(x) - F(p) - dF,(u) = R(x)(v, v). Then a : H,(I, R”) + Lz(H1(I, R”), H,(I, L”(R”, RP)))

is continuous by the first part of the theorem and if 0 and x = p + a are in H,(I, R”)

F(x) - F(5) - dF,(p) = @x)(p, p). It follows that F is differentiable at 0 and dF,, = dF,.

Since z, is a continuous function of 5 by the first part of the lemma F is C’.

q.e.d.

The followitig is trivial:

THEOREM (5). Consider R” and R” as complementary subspaces of RmCn. Then the map

(A, 5) -+ II + 5 is an isometry of H,(I, R”) @ H,(I, R”) onto H,(Z, Rm+“).

DEFINITION. If V is afinite dimensional Cl-manifold we define H,(I, V) to be the set of

continuous maps 5 of I into Vsuch that p D 5 is absolutely continuous and I[(rp 0 a)'[] locally

square summable for each chart rp for V. If V is C2 and 5 E H,(Z, V) we define H,(I, V), =

(A E H,(I, T(V))ll(t) E Va(lj for all t E Z}. If P, Q E V we define R(V; P, Q) =

{a E H,(Z, V)la(O) = P, a(l) = Q} and if 5 E R(V; P, Q) we define f2(V; P, Q), =


{A E H,(I, V),lA(O) = 0, and L(1) = O,}. We note that H,(f, V), is u rector space under

pointwise operations and thot fl( V; P, Q), is a subspace of H,(I, V),.

THEOREM (6). If V is u closed C”” -submanifold of R” (k 2 1) then H,(I, V) consists

of all a E H,(I, R”) such that a(I) c V and is a closed Ck-submanifold of the Hilbert space

H,(I, R”). IfP, Qe VthenR(V; P, Q)isaclosedC’-submanifoldofH,(/ V). IfaE H,(I, V)

then the tangent space to H,(Z, V) at a (us a submanifold of H,(J, R”)) is just H,(f, V),

which is equal to (1. E H,(I, R”)lA(t) E VS(,) t E I} and if G E fl( V; P, Q) then the tangent space

to fZ( V; P, Q) at a is just n( V; P, Q), which equals {i E H,(J, V),(i.(O) = A(1) = O}.

Proof. That H,(Z, V) equals the set of a E H,(/, R”) such that a(l) E V is clear, and

so is the fact that H,(f, V), and Q( V; P, Q), are what they are claimed to be. Since V is

closed in R” it follows that H,(Z, V) is closed in C’(/, R”), hence in H,(I, R”) by Corollary (2)

of Theorem (3). In the same way we see that !2(V; P, Q) is closed in H,(J, R”) and that

H,(I, V), and a( V; P, Q), are closed subspaces of H,(J, R”). Since V is a C’+‘-submanifold

of R” we can find a C’+‘-Riemannian metric for R” such that V is a totally geodesic sub-

manifold. Then if E: R” x R” -+ R” is the corresponding exponential map (i.e.

t + E(p, ta) is the geodesic starting from p with tangent vector L.), E is a CkC2-map. Let

a E H,(J, V) and define rp : H,(I, R”) + H,(J, R”) by q(l)(t) = E(a(t), l(t)). Then by

Theorems (4) and (5) cp is Ck and clearly ~(0) = 6. Moreover by Theorem (4) dq,(l)(t) =

dE$‘)(I.(t)) where I?(‘) (u) = E(a(t), c). By a basic property of exponential maps dE,““’ is

the identity map of R”, hence dq, is the identity map of H,(J, R”) so by the inverse function

theorem q maps a neighborhood of zero in H,(J, R”) CL-isomorphically onto a neighbor-

hood of c in H,(I, R”). Since V is totally geodesic it follows that for i near zero in H,(I, R”),

~(2) E H,(I, V) if and only if 1 E H,(I, V), and similarly if a E l2(V; P, Q) then

rp(A> E n(V; P, Q) if and only if A E S2( V; P, Q),. Consequently cp-’ restricted to a neighbor-

hood of a in H,(I, V) (respectively Q( V; P, Q)) is a chart in H,(Z, V) (respectively

NV; P, Q)) which is the restriction of a CL-chart for H,(I, R”), so by definition H,(I, V)

and fI( V; P, Q) are closed CL-submanifolds of H,(J, R”) and their tangent spaces at CT are

respectively H,(I, V), and Q( V; P, Q),. -_

q.e.d.

THEOREM (7). Let V and W be closed Ckc 4 -submanifolds of R” and R” respectively

(k 2 1) and let cp : V + W be a Ck+4-map. Then @ : H,(I, V) -+ H,(I, W) dejned by

@(a) = rp o a is a CL-map of H,(I, V) into H,(f, W). Moreocer d@, : H,(I, V), -+

HlU9 W)+,) is given by G,@)(t) = dpScl,(l.(t)).

Proof. By’s well-known theorem of elementary differential topology cp can be extended

to a Ck+4-map of R” into R” and Theorem (7) then follows from Theorems (4) and (6).

DEFINITION. Let V be a Cke4 -manlyold of finite dimension (k 1 1) and let j : V -+ R”

be a Ck+4 -imbedding of V as a closed subman!folbld of a Euclidean space (such always exists

by a theorem of Whitney). Then by Theorem (7) the CL-structures induced on H,(J, V)

and n( V; P, Q) as closed C’-submanifolds of H,(I, R”) are independent of j. Henceforth

we shall regard H,(Z, V) and f2(V; P, Q) as Ck-HiIbert manifolds. Jf cp : V -+ W is a Ck+4-

map then by Theorem (7) @ : H1(I, V) -+ H,(I, W) defined by @(a) = rp D u is a Ck-map and

d@&)(t) = dqo,(&(t)). W e note that @ maps fl( V; P, Q) C’ into fl( W; q(P), p(Q)).

326 RICHARDS. PALAIS

THEOREM (8). The function V + H,(I, V), p + (p is a jiunctor from the category of

finite dimensional C” 4 -mantfohis to the category of CC-Hilbert mantfohis (k L 1).

DEFINITION. Let V be a C’+4 -finite dimensional Riemannian manifold (k L: 1). We

define a real valuedfunction Jv on H,(I, V) called the action integral by

J’(a) = i I

’ /a’(r)il’ dt. 0

THEOREM (9). Let V and W be Ck+4 -Riemannian manifolds of finite dimension and let

cp : V 4 W be a Ckf4-local isometry. Then Jv = Jw O q,

Proof. g(a)‘(t) = (cp O a)‘(t) = dp,(,,(a’(t)). Since dqacr, maps Vfl/a(,i isometrically into

U’,p(a(,)ir j@(a)‘(t)~l = Qa'(t) 11 and the theorem follows.

COROLLARY (1). If V is a Ck+4- Riemannian submanifold of the Ck’4-Riemannian

manifold W then Jv = J”IH,(I, V).

COROLLARV (2). If V is a closed CL’” -submantfold of R” then J’(a) = fl,Lall$

Consequently Jv : H,(I, V) --) R is a CL-map.

Proof. By definition JR”(o) = $l\Lul!& so the first statement follows. Since JR” is a

continuous quadratic form on H,(I, R”) (Theorem (I)), J’” is a C”-map of H,(I, R”) into

R, hence its restriction to the closed CL-submanifold H,(f, V) is C’.

q.e.d.

COROLLARY (3). If V is a complete finite dimensional CkC4-Riemannian mantfold then

Jv is a CL-real valuedfunction on H,(Z, V).

Proof. By a theorem of Nash [7] V can be Cki4 -imbedded isometrically in some R”,

so Corollary (3) follows from Corollary (2).

Remark. Let W be a complete Riemannian manifold, V a closed submanifold of Wand

give V the Riemannian structure induced from W. Let pv and pw denote the Riemannian

metrics on V and W. Then clearly if p, q E Vp,(p, q) 1 pw(p, q) since the right hand-side

is by definition an Inf over a larger set than the left. Hence if {p,) is a Cauchy sequence in

V it is Cauchy in Wand hence convergent in Wand therefore in V because V is closed in W.

Hence V is complete. From this we see that

THEOREM (10). If V is a closed Ck+4 -submanifold of R” then H,(I, V) is a complete

C’- Riemannian manifold in the Riemannian structure induced on it as a closed Ck-submantfold

of H,(I, R”).

Caution. The Riemannian structure on H,(Z, V) induced on it by an imbedding onto

a closed submanifold of some R” depends on the imbedding. To be more precise if V and

W are closed submanifolds of Euclidean spaces and q : V + W is an isometry it does not

follow that (p : H,(Z, V) + H,(Z, W) is an isometry. It seems reasonable to conjecture that

(p is uniformly continuous but I do not know if this is true.

THEOREM (11). If V is a closed Ck+4 -submantfold of R” and P, Q E V then !A( V; P, Q)

is included in a translate of H:(I, R”), and NV; P, Q), c H:(I, R”).


Proof. If a and p are in f2(V; P, Q) then (a - p)(O) = P - P = 0 and (a - p)(l) =

Q - Q = 0, and the first statement follows. The second statement is of course a con-

sequence of the first, but it is also a direct consequence of the definition of fI( V; P, Q),.

COROLLARY (1). If we regard R(V;P, Q) as a Riemannian submanlfold of H,(I. R”)

then the inner product (,), in S2( V; P, Q)# is given by (p, A), = (Lp. LA),.

Proof. Immediate from Theorem (1).

COROLLARY(~). If.S E Q(V; P, Q>undifJ ’ is bounded on S then S is rota& bounded

in @)(I, R”) and H,(I, R”).

Proof. Since J’(a) = +ljLolli (Corollary (2) of Theorem (9)) &,I, is bounded on S.

Since S is included in a translate of H:(I, R”) it follows from Theorem (1) that S is bounded

in H,(I, R”), hence by Corollary (2) of Theorem (3) S is totally bounded in CotI. R”) and

Ho(4 R”).

COROLLARY (3). If {on} is a sequence in fl(V; P, Q) and ,‘L(o,, - a,);,, -, 0 as

n, m + co then 0, concerges in a( V; P, Q).

ProoJ By Theorem (11) u, - 6, E H:(I, R”) hence by Theorem (1) {a,} is Cauchy

in H,(I, R”), hence convergent in H,(I, R”). Since R(V; P, Q) is closed in H,(Z, R”) the

corollary follows.

DEFINITION. Let V be u closed CL+4 -submanifold of R” (k > 1) and let P, Q E V. If

0 E fl(V; P, Q) we define h(u) to be the orthogonal projection of La on the orthogonal

complement ofL(R(V; P, Q),) in H,(I, R”).

THEOREM (12). Let V be a closed Ck+4 -submanifold of R” (k 2 l), P, Q E V and let J

be the restriction of J” to n( V; P, Q). lf we consider a( V; P, Q) as a Riemanniun manifold

in the structure induced on it as a closed submanifold of H,(I, R”), then for each IJ E fl( V: P, Q)

VJ, can be characterized as the unique element of fl( V; P, Q), mapped by L onto La --h(o).

Moreover l[VJ..II, = IlLa - h(a)II,.

Proof. Since fl(V; P, Q), is a closed subspace of H,(/, R”) (Theorem (6)) and is

included in H:(I, R”) (Theorem (11)) it follows from Theorem (1) that L. maps S2( V; P, Q),

isometrically onto a closed subspace of H,(f, R”) which therefore is the orthogonal com-

plement of its orthogonal complement. Since La - h(u) is orthogonal to the orthogonal

complement of L(n(V; P, Q),) it is therefore of the form Li. for some I E fZ(V; P, Q),

and since L is an isometry on R(V; P, Q), 1 is unique and Ill/i, = iiLl)/, = l/La - h(a)/i,

so it will suffice, by the definition of VJ,, to prove that U.,(p) = (2, p), for p E fJ( V; P, Q),,

or by Corollary (1) of Theorem (ll), that dJ,@) = (Ll, Lp), = (L,, - h(u),Lp), for

p E n(V; P, Q),. Since by definition of h(u) we have (h(u), Lp), = 0 for p E fI( V; P, Q),

we must prove that U,(p) = (La, Lp), for p E n(V; P, Q),. Now JR”(a) = +l/Lajlg

(Corollary (2) of Theorem (9)) so d-I?(p) = <La, Lp), for p E H,(I, R’). Since

J = JR”IR( V; P, Q) by Corollary (1) of Theorem (9) it follows that dJ, = dJyla( V; P, Q),.

q.e.d


$14. VERIFICATION OF CONDITION (C) FOR THE ACTION IhTEGRAL

In this section we assume that Y is a closed CLA4 -submanifold of R” (k >_ 3). P, Q E V

and J = J”IR( V; P, Q). We recall from the preceding section that S2( V; P, Q) is a complete

Ck-Riemannian manifold in the Riemannian structure induced on it as a closed submanifold

of H,(f. R”) and J is a Ck-real valued function. Our goal in this section is to identify the

critical points of J as those elements of n( V; P. Q) which are geodesics of V parameterized

proportionally to arc length, and secondly to prove that J satisfies condition (C).

DEFINITION. We define a CkC3-map R : V + L(R”, R”) by R(p) = orthogonal projection

qf‘ R” on V,. If 0 E fi( V; P, Q) we define Ti( V; P, Q), to be the closure of 52( V; P, Q), in

H,( I. R”) and we define P, to be the orthogonal projection of H,(I, R”) on Ti( V; P, Q),.

THEOREM (I). If G E R(V; P, Q) then a( V; P, Q), = {j. E H,(I, R”)/E.(t) E Vo(,, for

almost all t E I} and if i. E H,(I, R”) rhen (Psi,)(t) = fi(a(t))E.(t).

Proof. Define a linear transformation rr, on H,(I, R”) by (Q.)(t) = i2(g(t))A(t).

Since Q(a(t)) is an orthogonal projection in L(R”, R”) for each t E I it follows from the

definition of the inner product in H,(Z. R”) that T[, is an orthogonal projection. From the

characterization of O(V; P, Q), in Theorem (6) of 513 it is clear that II, maps HT(f, R”)

onto R(V; P, Q),. Since H:(f, R”) is dense in H,(I, R”) it follows that the range of rr,

is a( V; P, Q),, hence n, = P,. On the other hand it is clear that i E H,(I, R”) is left fixed

by rt, if and only if E.(t) E V,,(,, for almost all t E 1. Since the range of a projection is its

set of fixed points this proves the theorem.

q.e.d.

COROLLARY (I). If d E !2( V; P, Q) then

P,,(H,(L R”)) = H,(& V),

and

P,(H :(I, R”)) = !2( V; P, Q),. -_

COROLLARY (2). If d E Q( V; P, Q) then P,L,a = Lg.

Proof. Clearly (La)(t) = a’(t) E Vs/a(rl whenever a’(t) is defined, so La E a( V; P, Q),.

THEOREM (2). Let T E H,(f, L(R”, RP)) and define for each J. E H,(I, R”) a measureable

function T(2) : I + RP by T(l)(t) = T(t)l.(t). Then:

(1) T is a bounded linear transformation of H,(I, R”) into L’(I, RP);

(2) IfTand E. are absolutely continuous then so is TZ. and (TA)‘(t) = T’(t)A(t) + T(t)].‘(t);

(3) Jf TE H,(I, L(R”, Rp)), 1 E H,(I, R”) then Tl. E H,(Z, RP).

Proof. If n = p = 1 then (1) follows from Schwartz’s inequality, (2) is just the product

rule for differentiation and (3) is an immediate consequence of (2). The general case

follows from this case by choosing bases for R” and RP and looking at components.

DEFINITION. Given Q E t2( V; P, Q) we define G, E H,(I, L(R”, R”)) by G, = R D Q and

we define Q, E H,(f, L(R”, R”)) by Q, = G,‘.


Remark. That G, E H,(I, UR”, R”)) follows from Theorem (4) of Section (13).

THEOREM (3). Let CJ E R(V; P, Q). Ifp E H,(Z, R”) then (LP, - P,L)p(t) = Q.(r)p(r)

Girenfe H,(I, R”) define an absolutely continuous map g : I + R” by

Then lfp E HT(Z, R”)

g(r) = ‘Q,W/W ds. s 0

(L (LPO - P.J)P)o = (97 -LP)o.

Proof. Since P,p(t) = G,(r)p(t) and P,(Lp)(t) = G,,(t)p’(t) by Theorem (1). the fact

that (LP, - P,L)p(t) = Q,(t)p(f) is an immediate consequence of (2) of Theorem (2).

By (1) of Theorem (2) s -+ QJs)f( s 1s summable so g is absolutely continuous. Next note ) .

that since G,(t) = n(a(t)) IS self-adjoint for all t. Q,(I) = G,‘(r) is self-adjoint wherever

it is defined, hence

<f, (LP, - P&h), = s

lUiO. Q,(MO) tit = l<Q,(OfOL ~(0) dt 0 J’ 0

= ‘<s’W, p(G) cit.

l 0

Then if p E H:(I, R”) Theorem (2) of $13 gives

if7 (LP, - PAP), = (Y7 -LP)o q.e.d.

We now recall that if 0 E O( Y; P, Q) then in $13 we defined h(a) to be the orthogonal

projection of La on the orthogonal complement of L(!2( V; P, Q),) in H,(I, R”). By

Corollary (1) of Theorem (1) above it follows that (h(a), LP,p) = 0 if p E H:(I, R”).

THEOREM (4). If0 E n( V; P, Q) then P,h(a) is absolurely continuous and (P,h(a))‘(t) =

QAt)h(a)(t).

Proof. If p E H*(I, R”) then --

VA(a), LP), = (h(a), P,Lp)o = (h(a), (P& - LP,)p)o

since (h(a), LP,,p) = 0. Hence by Theorem (3) (P,h(a), Lp), = (g, Lp), if we define g

to be the absolutely continuous map of I + R”

s(l) = ‘QhVi4(4 ds. s 0

Then P,h(a) i g is orthogonal to L(H:(I, R”)) so by Theorem (1) of 913 P,,h(a) - g is

constant. Since g is absolutely continuous so is P,h(a) and they have the same derivative.

But g’(t) = Q,Cr)h(a)(t>.

q.e.d.

THEOREM (5). If Q is a criticalpoint ofJthen CTE Ck+4 (1,-V) andmoreocer d is ecerywhere

orthogonal to V. Concersely giren a E fl( V; P, Q) such that a’ is absolutely continuous and

(a’)’ is almost eterywhere orthogonal to V, a is a critical point of J.

Proof. By Theorem (12) of $13 if 0 is a critical point of J then La = h(a). Since


P,Lo = Lo (Corollary (2) of Theorem (1) above) it follows that P,h(a) = h(a) so by

Theorem (4) 6’ is absolutely continuous (so G is C’) and

(*) U”(f) = Q,(r)a’(t).

Now since R : Y -+ L(R”, R”) is Ckc3 and

Q&t) = ; Q(@G)

it follows that if D is Cm (1 I m < k + 3) then Q,(t) is Cm-‘, hence by (*) (T” is C”- ’

so CT is C’“’ *. Since we already know CJ is C’ we have a start for an induction that gives

u E CL”. If p E n(V; P, Q), then La = h(a) is orthogonal to Lp, so by Theorem (2) of

$13 (and the fact that fl(Y; P, Q), E H:(I, R”)-Theorem (11) of 913) 0” is orthogonal

to p. Since (T” and p are continuous it follows that (a”(r), p(t)) = 0 for all t E 1. Now it

is clear that if t E I is not an endpoint of I and z’,, E V,,(rj then there exists p E n(V; P, Q)n

such that p(t) = L‘~, hence o”(t) is orthogonal to VU(tj, and by continuity this holds at the

endpoints of I also. Conversely suppose .d E fi(V; P, Q) is such that 0’ is absolutely

continuous and a”(f) is orthogonal to Va,,, for almost all t E I. Then by Theorem (2) of

$13 La is orthogonal to L(n(V; P, Q),) so La = h(a) and by Theorem (I 2) of $13 CJ is a

critical point of J.

q.e.d.

COROLLARY. If c~ E fZ( V; P, Q) then CJ is a critical point of J if and only if a is a geodesic

of V parameterized proportionally to arc length.

Proof. It is a well-known result of elementary differential geometry that G E C’(f, V)

is a geodesic of V parameterized proportionally to arc length if and only if 6” is everywhere

orthogonal to Vt

LEMMA. Giren a compact subset A of V there is a constant K such that

I

1

llQ,~MOil dt I ~IiLdiOilpilO 0

-_

for allp E H,(I, R”) and aN a E H,(I, R”) such that a(Z) -C A.

Proof. Let A* be the compact subset of R” x R” x R” consisting of triples (p, c, x)

such that p E A, r is a unit vector in V, and x is a unit vector in R”. Since R is Ck+3,

(p, L’, x) + j!dR,(r)xll is continuous on A* and hence bounded by some constant K. Since

Q.AO = 2 G,(O = $&JO)) = dR,&‘(t))

it follows that

IIQ,WPOJII 5 Klb’O)il~ ll~v)ll. Integrating and applying Schwartz’s inequality gives the desired inequality.

q.e.d.

We now come to the proof of condition (C).

t Seaz EISENHART: An introduction to Differential Geometry, p. 246


THEOREM (6). Let S G f2( Y; P, Q) and suppose J is bounded on S but that ,,VJ;I is not

hounded away from zero on S. Then there is a critical point of J adherent to S.

Proof. By Theorem (12) of $13 we can choose a sequence {g,,j in S such that

\lVJ,“j/ = JILo, - h(a,)i!, + 0. Since each P,” is a projection, hence norm decreasing. it

follows from Corollary (2) of Theorem (1) of this section that ;iLg, - P,_h(a,) :,, -+ 0, and

by Corollary (2) of Theorem (1 l), $13, we can assume that I;gn - orni = -+ 0 as m, n -+ a~.

It will suffice to prove that :IL(a, - a&i,, + 0 as m, n + o for then by Corollary (3)

of Theorem (1 l), $13, 6, will converge in (fiv; P, Q) to a point g in the closure of S, and

since ‘iVJ;I is continuous it will follow that i!VJ, i = 0, i.e. 0 is a critical point of J. But

iiL(a. - 0,)/l: = (Lo-,, Lia, - ffm)jO - <Lo,. L(cT, - a,ljo

hence it will in turn suffice to prove that (Lo,, L(a, - CT,)), --t 0 as m, n -+ 30. Now

;jLlS’i2 = 2J(a,) is bounded, hence l\L(cr, - a,)/,, is bounded, and since L,_ - P,_h(cs,)

+ 0 in H,(Z, R”) it will suffice to prove that (P,,“/r(cr,) .L(c, - a,)), + 0 as tn. n -+ co.

Recalling that 0, - CJ,,, E H:(I, R”) (Theorem (I 1) of 913) it follows from Theorem (4)

above and Theorem (2) of $13 that

Jo

and since /Iu, - ornlln -+ 0 it will suffice to prove that

s ; I, Qo.(M~,)(~) II dt

is bounded. Let A be a compact set such that a,(J) c A

the fact that {a.} being uniformly Cauchy is uniformly

exists K such that f’

(the existence of A follows from

bounded). By the lemma there

-_

J IIQJ0h(~,)(~>li dt I KllL~,liollN~,)llo. 0

Now it has already been noted that IILa,jl, is bounded, and since

is j:h(o,) Ilo.

ilLa, - hi:, --, 0 so

q.e.d.

For the sake of completeness we give here a brief description of the classical conditions

that the critical points of J be non-degenerate and of a geometrical form of the Morse

Index Theoremt.

Let E denote the exponential map of V, into Y; i.e. if z: E V, then E(t.) = a( i/nl/)

where cr is the geodesic starting from P with tangent vector r*//ir’/. Then E is a Ck+2-map.

Given u E V, define n(v) = dimension of null-space of dE,. If I.(c) > 0 we call u a conjugate

vector at P. A point of V is called a conjugate point of P if it is in the image under E of

t For a detailed exposition the reader is referred to I. M. Singer’s Nofes on Di’rentiol Geometrv (Mimeographed, Massachusetts Institute of Technology. 1962).

332 RICHARD S. PALAiS

the P. By an easy special case of Sard’s Theoremt the set of

conjugate points of P has measure zero and in particular is nowhere dense in V.

Given u E E-‘(Q) define i; E n( V; P, Q) by t’(t) = E(t(c)). Then i; is a geodesic

parameterized proportionally to arc length (the proportionality factor being ;$;I), hence a

critical point of J by corollary of Theorem (5>, and conversely by the same corollary any

critical point of J is of the form v’ for a unique tr E E-‘(Q).

Non-degeneracy Theorem

If v E E-‘(Q) then ii is a degenerate point of J and only if is a

cector P, hence has only non-degenerate critical points lf and only if Q is not a conjugate

point of P. Itfollolvs that if Q is chosen outside a set of measure zero in V then J : fl( V; P, Q)

+ R has only non-degenerate critical points.

Morse Index Theorem

Let v E E- '(Q). Then there are only a finite number off sarisfying 0 < t < 1 such that

tr is a conjugate vector at P and the index of i; = 1 A(tv). In particular each critical point o<r< 1

of J : !2( V; P, Q) + R has finite index.

$15. TOPOLOGICAL IMPLICATIONS

Until now we have given no indication of why one would like to prove theorems such

as the Main Theorem. Roughly speaking the answer is that as a consequence of the Main

Theorem one is able to derive inequalities relating the number of critical points of a given

index with certain topological invariants of the manifold on which the function is defined.

These are the famous Morse Inequalities and are useful read in either direction. That is,

if we know certain facts about the topology of the manifold they imply existential statements

about critical points, and conversely if we know certain facts about the critical point

structure we can deduce that the topology of the manifold can be only so compli-

cated.

As a start in this direction we will show that if M is a complete C2-Riemannian

manifold and f: M + R is a C2-function bounded below and satisfying condition (C) then

on each component of Mf assumes its lower bound. Note that we do not assume that the

critical points off are non-degenerate, however since it is clear that a point where f assumes

a local minimum is a critical point, and is of index zero if non-degenerate, it follows that

if the critical points off are all non-degenerate then there are at least as many critical points

of index zero as there are components of M. This is the first Morse inequality.

In what follows we denote the frontier of a set K by R.

THEOREM (1). Let M be a connected Cl-manifoldf: M + R a non-constant C’-function

and K the set of critical points ofJ Then f(K) = f(k).

t DE RHAM: Variete’s Difirentiable, p. 10.


Proof. Let p E K. We will find x E k such that f(x) = f(J). Choose q E M with

.fiq) #f(p) and G : I + M a Cl-path such that a(0) = p and a( 1) = q and let g(t) = f(b(t)).

Then g’(t) = dfec,,(b’(r)) and since g is not constant. g’ is not identically zero, so a(I) is

not included in K. Let t, = Inf{t E Ila(t) $ Ki. Then x = o(r,) E k and since g’(r) = 0

for 0 I I I fO.f(s) = g(LO) = g(0) = f(p).

q.e.d.

THEOREM (2). Let M be u Cl-Riemanniun manifold, f: M + R a C’-function

satisfving condition (C) and K the set of critical points of f. Then f Ik is proper;

i.e. gicen - x < a < h < x. h; n f - ‘([a, b]) is compact (note rcle do not assume that M

is complete).

Proof. Let (p,) be a sequence in ti with II I .fp,) 5 6. Since h’ is closed it will suffice

to prove that {p,) has a convergent subsequence. Since p, E fi we can choose q. $ K

arbitrarily close to p,. In particular since ;Vf is continuous and ~ Vf,,i! = 0 we can

choose q, so close to p, that

llVfb.ll < d 7 a - 1 <j(qn) < b + I

and also

1 P(4.* P”) < ;

where p is the Riemannian metric for M. Then by condition (C) a subsequence of {q.} will

converge to a critical point p off. Since

the corresponding subsequence of {II,) will also converge to p. q.e.d.

Remark. f IK need not be proper-for example if M is not compact and f is constant

then f trivially satisfies condition (C) and K = M. -_

THEOREM (3)t. Let M be a complete C”-Riemannian manifold without boundary,

f: M + R a C2-function satisfying condition (C) and 0 : (r, ,Ll) - M a maximum integral

curre of VJ Then either lim f(a(t)) = co or else /I = 00 and D(I) has a critical point off

as a limit point as t --+ co. Similarly either lim f(o(t)) = -co or else z = -m and o(t)

has a critical point off as limit point as t --t - SO.

Proof. This is just the lemma to Proposition (2) of $10 restated verbatim. We simply

note that in the proof of that lemma we did not use the standing assumptions of $10 that f

was C3 or that the critical points off were non-degenerate.

THEOREM (4). Let M be a complete C2-Riemannian man[fold and f: M -+ R a C2-

function satisfying condition (C). [ff b IS ounded below on a component M, of M then f IM,

assumes its greatest lower bound.

t In this regard see also Proposition (3) of $12.


Proof. We can assume that A4 is connected. Let B = Inf{f(x)l.r E ,t!). Given E > 0

choose p E M such that_@) < B + E. If d : (z, /?) 4 M is the maximum integral curve of

Vfwith initial condition p then by Theorem (3) r = - x and ~$1) has a critical point q as

limit point as t -+ -co. Since f(a(t)) is monotonic non-decreasingf(:q) < B + E. Since

the theorem is trivial iffis constant we can assume f is not constant and it follows from

Theorem (1) that if K is the set of critical points off we can find x in E(‘, the frontier of K,

such thatf(x) < .B + E. Choose x, E R such that

Blf(X,) < B + ‘. n

Then by Theorem (2) a subsequence of {x,1 will converge to a point x and clearly,f(x) = B.

q.e.d.

COROLLARY (1). If the set of critical points off has no interior and iff’is hounded be/o,\*

an ,bf then f assumes its greatest IoH-er bound.

Proof. If B is the greatest lower bound off then for every positive integer n we can

choose x, E h; (a minimum off on some component of M) such that

Since K has no interior and is closed K = R, so by Theorem (2) a subsequence of {x,} will

converge to a point x wheref(x) = B.

q.e.d.

COROLLARY (2). If V is a C6 complete Riemannian man/fold and P. Q E V then [he

action integral J” assumes its greatest lolc*er bound on each component qf R(V; P, Q) and

also on i2( V; P, Q).

Proof. We saw in Theorem (6) of 914 that condition (C) is satisfied. If K is the set of

critical points of J”IR( V; P, Q) then by the corollary of Theorem (5) of $14 the elements c

of K are (geodesics) parameterized proportionally to arc length. By making a small para-

meter change we can get element of R(V; P, Q) arbitrarily close to 0 which are not

parameterized proportionally to arc length, hence K has no interior.

Remark. If V is a complete Riemannian manifold and P, Q E V then given an abso-

lutely continuous path 0 : I + I/ with a(O) = P, a(l) = Q define the length of 6, L(U), by

L(c) = iijcr’(l)ii dt. s 0

Then by Schwartz’s inequality if G E fi( V; P, Q) L(O) I (W(G))“’ and moreover equality

occurs if and only if 116 /I is constant, i.e. if and only if 0 is parameterized proportionally to

arc length. Now if CT : I --+ V is absolutely continuous and a(0) = P, o(l) = Q we can

reparameterize d proportionally to arc length, getting y : I -+ V. Then y E 0( V; P, Q) and since

arc length is independent of parameterization L(y) = L(a). Since reparameterizing also does

not affect the homotopy class of u we see that if J” assumes its greatest lower bound on a

component of fi( V; P, Q) at a point y (SO that y is a geodesic parameterized proportionally

MORSE THEORY ON HILBERT MAMFOLDS 335

to arc length) then among all absolutely continuous paths joining P to Q and homotopic

to 7, 7 has the smallest length. Together with the preceding corollary this gives:

THEOREM (5). If V is a complete C6-Riemannian manifald. P, Q E V then gicen an>

homotop>, class of paths joining P E Q there is a geodesic in this class rc,hose length is less than

or equal to that of any other absolutely continuous path in the class. Moreowr there is a

geodesic joining P to Q rc,hose length is p(p, q).

Let Hi be a Hilbert space of dimension di. i = 1, . . . . n, D, the closed unit disc in Hi and

Si the unit sphere in Hi. Let gi : Si -+ X be continuous maps with disjoint images in a

topological space X. We form a new space X ug, D, u . up, D, (called the result of

attaching cells of dimension d,, ,.., d, to X by attaching maps gr, . . . . gn) by taking the

topological sum of X and the Di and identifying y E Di with gi(y) E X. Suppose now that

di < w i = I. . . . . m and di = m i > m. Then X ug, D, u . ug, D, is a strdng de-

formation retract of X u,, D, u . . . uy, D,. It will suffice to prove that if D is the unit

disc and S the unit sphere in a Hilbert space H of infinite dimension then S is a strong

deformation retract of D, or since D is convex it will suffice to find a retraction p : D + S.

By a theorem of Klee [2.2 of 31 there is a fixed point free map h : D -+ D (to see this note

that if {.Y,:-,~~ is a complete orthonormal basis for H then

f(f) = (

cos VI,. + (sin v)x”+, ,IIt<!lfl

defines a topological embedding of R onto a closed subset F of D. Since F is an absolute

retract the fixed point free map f(t) -f(t + 1) of F into F can be extended to a map

h : D + F which is clearly fixed point free). We define p : D + S by p(x) = point where the

directed line segment from h(x) to .Y meets S.

It now follows (by excision) that if H, denotes the singular homology functor with any coefficient group G then

H,(A’ u,, D, u ug,, D,. A’) z ic,H,(D”z. Sd8-‘)

hence for any positive integer r

H,(X ug, D, u . . . u,,, D,, X) = G p’r’

--

where p(r) is the number of indices i = I. _.,, n such that di = r.

Next let N be a Hilbert manifold with boundary and suppose M arises from N by

disjoint C-attachments (Jr, . . . . f,) of handles of type (d,, el), . . . . (d,, e,) ($11). Define

attaching maps gi : Sdi- ’ -+ dN by gi(y) = fi(y, 0). (Note that since f, : Ddi x D” -+ M

is a homeomorphism each gi is a homeomorphism.) Then clearly N ufr( Ddl x 0) u . . ufXDd' x 0) can be identified with N ug, D”’ u . ugn Dd”. We shall now prove that

Nu ;/,(D”%O) i=l

is a strong deformation retract of M, hence by what we have just proved above that if

di < 00 i = 1, . . . . m, di - CO i > m then Nu,, Dd’ u . . . ug Dd” is a strong deformation


retract of M. It will suffice to prove that (od x 0) u (Sd-’ x D’) is a strong deformation

retract of D“ x D’, and since D“ x D’ is convex it will suffice to define a retraction r of

D“ x D’ onto (od x 0) u (Sd-’ x 0’). Deiine r(x, 0) = (x, 0) and if y # 0 define

dx,_V)= (j-$-jjj,O) if l/xl/ 51 -y

4x,L’)= ( & WI + /IY// - 2) & ) if Ijx// 2 1 -u 2 .

From the above remarks together with the theorem of $12 we deduce:

THEOREM (6). Let M be a complete C3-Riemannian manifold, f: M -+ R a C3-function

satisfying condition (C) all of whose critical points are non-degenerate, c a critical value off.

pl, . ., pn the critical points of finite index on the level c, and let di be the index of pi. If

c is the only critical tlalue off in a closed interral [a, b] then M, has as a deformation retract

M, with cells of dimension d,, . .., d, disjointly attached to S,M, by homeomorphisms of the

boundary spheres. Hence if Hk denotes fhe singular homology funcfor in dimension k with

coeficient group G then H,(M,, M,) z G”” where C(k) is the number of critical points of

index k on the lecel c.

Remark. The surprising fact about Theorem (6) is that the homotopy type of(M,, M,)

depends only on the critical points of finite index on the level c, those of infinite index being

homotopically invisible. This is of course just a reflexion of the theorem of Klee that the

unit disc modulo its boundary in an infinite dimensional Hilbert space is homotopically

trivial. If it were not for this unexpected phenomenon we would have to make the rather

unaesthetic assumption that all critical points were of finite index in order to derive Morse

Inequalities.

In deriving the Morse Inequalities we shall follow Milnor closely. Let F denote a

fixed field and H, the singular homology functor with coefficients F. We call a pair of

spaces (X, Y) admissible if H,(X, Y) is of finite type, i.e. each H,(X, Y) is finite dimensional

and H,(X, Y) = 0 except for finitely many k. From the exact homology sequence of a

triple (X, Y, 2) it follows that if (X, Y) and ( Y, 2) are admissible then so is (A’, Z). We

call an integer valued function S on admissible pairs subadditive if %A’, Z) I S(X, Y) +

S( Y, Z) for all triples (X, Y, Z) such that (X, Y) and ( Y, Z) are admissible, and S is called

additive if equality always holds in the above inequality. Then by an easy induction if

X,, 2 A’,_, 2 . . . 2 A’, and each (Xi+r, Xi) is admissible it follows that (X,, X,-J is

admissible and n--l

S(x,, x0) 5 1 S(xi+ 1, xi) i=O

if S is subadditive, equality holding if S is additive.

DEFINITION. For each non-negative integer k we define in!eger ralued functions R, and

S, on admissible pairs by R,(X, Y) = dim Hk(X, Y) and

S,(X, Y) = c (- l)k_“R,(X, Y). mzzk


We define the Euler characteristic x for admissible pairs bj

x(X, Y) = f (- l)“R,(X, Y). m=O

LEMMA. R, and S, are subadditire and x is additire.

Proof. Let (X, Y, Z) be a triple of spaces such that (X, Y) and (Y, Z) are admissible.

From the long exact homology sequence of the triple (X, Y, Z)

i, im + H,( Y, Z) + H,(X, 2) + H,(X-, Y) “1 H,_ L( Y, Z) -+

we derive the usual three short exact sequences

O+im(~,+,)-+H,(Y, Z)-im(i,) -+O

O-+ im(i,) - H,(X, Z) ---) im(j,) -+ 0

0 --+ im(j,) - H,(X, I’)-im(ci,)-+O

from which follow

R,(Y, Z) = dim H,(Y, Z) = dim im(d,+ ,) + dim im(i,)

R,(X, Z) = dim im(i,) + dim im(j,)

R,(X, Y) = dim im(j,) +.dim imid,,,) hence

(,) R,(X, Z) - R,(X, Y) - R,(Y, Z) = -(dim im(d,) + dim im(a,+,)).

If we multiply (,) by (- l)k-m and sum over m from m = 0 to m = k we get

S,(X, Z) - S,(X, Y) - S,(Y, Z) = (-I)‘+’ dim im(S,) - dim im(d,+,)

which is negative since in fact do = 0. Similarly if we multiply (,) by (- I)” and sum over

all non-negative m we get x(X, Z) - z(X, Y) - x( Y, Z) = 0 since zk+ I = 0 for k

sufficiently large.

-q.e.d.

Now letfand A4 be as in Theorem (6). Let - co < a < b < CL) and suppose a and b

are regular values off: Let cr, . .., c, be the distinct critical values offin [a, b] in increasing

order. Choose a,, i = 0, . . . . nsothata = a, < c, < a1 < c2 < . . . < a,_, < c, < a, = b

and put Xi = M,, = {x E MIf(x) I a,}. Then by Theorem (6) (Xi+rr Xi) is admissible

and Rk(Xi+r, Xi) = number of critical points of index k on the level ci. Hence

sk(xi+l,~~i) =m~o(-l)k-m ( number of critical points of index m on level ci)

and

Axi+ 19 xi> = number of critical points of index m on the level ci).

Hence .

n-l

~osk(xi+l~ xJ =m~oC-l)k-m ( number of critical points of index m inf -‘([a, b])


while

n-1

i&oX(xi+ 13 xi) = f (- l)” ( number of critical points of index m in ~-‘([a, b]) m=O

Since Sk and x are subadditive and additive respectively we deduce

THEOREM (7). (MORSE INEQUALITIES.) Let M be a complete C’-Riemannian manifold,

f:M+RaC3-f t’ unc ran satisfying condition (C) all of whose critical points are non-degenerate.

Let a and b be regular values ofJ a < 6. For each non-negatire integer M let R, denote the

mth betti-number of (Mb, M,) relative to somejixedfield F and let C, denote the number of

critical points off of index m in f - ‘([a, b]). Then

R, 5 Co

and

R, - R, < C, - Co

,jo(- l)k-mR, 2 m$o(- I)‘-“&

x(kI,, M6) = f (- l)“R, = i (- l)“C,. m=O m=O

COROLLARY (1). R, I C, for all m.

COROLLARY (2). Iff is bounded below then the conclusions of the theorem and of Corollary

(1) remain valid if we interpret R, = mth betti-number of M, and C,,, = number of critical

points off having index m in Mh respecticely.

Proof. Choose a < glb J

COROLLARY (3). If f is bounded belorct then for each non-negatire integer m Rz I Cz,

where Rz is the mth betti-number of M and C,* is the total number of critical points off having

index m. (Of course either or both of Rz and C,* may be infinite.) -

Proof. By Corollary (2) we have C’z L R,(M,) for any regular value b off. Hence it

will suffice to show that if Rz = dim H,(M; F) 2 k for some non-negative integer k then

R,(M,) r k for some regular value b off. Let h,, . . . . h, be linearly independent elements

of H,,,(M; F), Z1, . . . . zk singular cycles of M which represent them, and C a compact set

containing the supports of z,, . . . . z,. Then as b + CO through regular values off the

interiors of the Mb form an increasing family of open sets which exhaust M, hence C c M,

for some regular value b of J Then zI, . . . . ;k are singular cycles of Mb, moreover no

non-trivial linear combination of them could be homologous to zero in M, since that same

combination would a fortiori be homologous to zero in M. Hence R,(M,) 2 k. q.e.d.

Caution. The assumption that f is bounded below is necessary in Corollary (3) as can

be seen by considering the identity map of R which has no critical points even though

R;(R) = 1.

We refer the reader to [8] for more delicate forms of the Morse Inequalities.

MORSE THEORY 0% HILBERT 41ASIFOLDS 339

Remark. If k’ is a complete C6-Riemannian manifold, P, Q E V define Q,,o( V) to be

the set of continuous maps cr : I + V such that a(O) = P and a(l) = Q, in the compact

open topology. The standard techniques of homotopy theory relate the topology of V and

that of fZ?,,,( V), while Theorem (7) and the results of $13 together give results concerning

the topology of n( V; P, Q). Clearly some sort of bridge theorem relating Q,,,(V) and

f2( k’; P, Q) is desirable. Now if V is imbedded as a closed submanifold of R” then

fZ( I’; P, Q) is a closed submanifold of H,(I, R”). While R,.o( k’) is a subspace of C”(Z, R”),

hence it follows from Corollary (2) of Theorem (3) ($13) that the inclusion map

i : C.l( b’; P, Q) 3 fi,,,( V) is continuous. The desired bridge theorem is the statement that

i is in fact a homotopy equivalence. A homotopy inverse can be constructed by using

smoothing operators of convolution type.

i:f !I: = s’i f(x) j2 du(x)

on the space CO(D”, R”) of CD-maps

( w h ere u is Lebesque measure on D”)

and

IlJiC = , EkllFfiiL I

Then the completion of C%(D”, R”) relative to the norm Ij IIL is a Hilbert space which we

denote by H,(D”, R”). We denote by H:(D”, R”) the closure in H,(D”, R”) of the set of

f in Cp( D”, R”) such that (WY)(x) = 0 if x E S”-’ and 121 I k - 1. Let V be closed

C”-submanifold of R” and let Hk( D”, V) = {f~ H,(D”, R”)IF(D”) c Y}. Ifg E H,.(D”, V) we define fY( V; g) = {f E H,(D”, V)jf - g E H:(D”, R”)}. It follows from the Sobolev

Inequalities that if 2k > n Hk( D”, V) and fY( V; g) are closed submanifolds of the Hilbert

space HK( D”, R”). More generally analogous Hilbert manifolds of HK maps of Miiito V’

can be constructed for any compact C” n-manifold with boundary M replacing D’. Note

that for k = II = 1, H,(D’, V) = H,(/, V) and fJ’(V,g) = n(V;g(O),g(l)). A question

that immediately presents itself is to find functions J : !A”( V; g) 4 R which are analogues

of the action integral and satisfy condition (C). If A is a strongly elliptic differential operator

of order 2k then J(f) = -) (AL f >. is such a good analogue of the action integral provided

_4f = 0 has no solutions f in Hc( D”, R”). In particular if L is an elliptic kth order elliptic

operator such that Lf = 0 has no solutions f in H:(D”, R”) then J(f) = +l/Lf 11; =+

(L*LJ, f ). is such a function (taking k = n = 1 and L = d/dt gives the ordinary action

integral). Smale has found an even wider class of functions which also satisfy condition (C).

Now let n < m and regard O(n) 5 O(m) in the standard way. Define an orthogonal

representation of O(n) on HJD”, R”) by (T/)(x) = T(f (T-*x)). If we take V = Sm-’ then

since V is invariant under O(n) it follows that H,(D”, V) is a invariant submanifold of

H,(D”, R”). Moreover if we define g E H,.(D”, V) by g(x) = (x,, . . . . x,, \/I - llx/!‘, 0 . . . 0)

then Tg = g for any T E O(n) and it follows that fY( k’, g) is also an invariant submanifold

330 RICHARD S. PALAlS

of H,(D”, Y), hence O(n) is a group of isometries of the complete Riemannian manifold

n”(Y, g). Now suppose A is a strongly elliptic differential operator of order 2k,

A : C”(D”, R”) -+ Cr(Dn, R”), such that A(Tf) = T(.4f) for all TE O(n), for example

A = Ak where A is the Laplacian

ii, 2.

Then J:R(V,g) 4 R defined by J(f) = +<Ai_f>,, satisfies J( 7f‘) = f for any T E O(U).

hence iffis a critical point of J so is rffor any TE O(n). and since non-degenerate critical

points of J are isolated, Tf = f if f is a non-degenerate critical point of J. But Tf = _/‘ is

equivalent to R(f(.r)) being a function F of ‘jxll where R is the distance measured along the

sphere .S”- ’ = V of a point on I/ to the north pole. Moreover F will satisfy an ordinary

differential equation of order 2k. With a little computation one should be able to compute

all the critical points and their indices and hence, via the Morse inequalities, get information

about the homology groups of f2”(V, g) (which has the homotopy type of the nth loop

space of .S”- ’ ). Clearly the same sort of process will work whenever we can force a large

degree of symmetry into the situation.

REFERENCFS

I. J. DIEUDONNE: Foundurions of Mondern Analysis. Academic Press, New York, 1960. 2. J. EELLS: On the geometry of function spaces, Symposium Inrernacional de Topolo,qia A!quhraica (,Me.riw.

1956), pp. 303-308. 1958. 3. V. KLEE: Some topological properties of convex sets, Trans. Amer. Marh. Sot. 78 (1955). 30-45. 4. S. LANG: Introduction to Differentiable Manifolds, Interscience, New York, 1962. 5. J. MILNOR: Morse theory, Ann. Math. Sud. No. 51, (1963). 6. M. MORSE: The calculus of variations in the large. Colloq. Lect. Amer. Math. Sot. 18 (1933). 7. J. NA.SH: The imbedding problem for Riemannian manifolds, Ann. Mafh., Princeton 63 (1956). 20-63 8. E. PITCHER: Inequalities of critical point theory, Bull. Amer. r2fanth. Sot. 64 (1958). I-30.

Brandeis Unirersity.

MORSE THEORY ON HILBERT ~ANrF~LDS - Richard Palaisvmm.math.uci.edu/PalaisPapers/MorseThOnHilbertManifolds.pdf · MORSE THEORY ON HILBERT ~ANrF~LDS ... M,) in terms of the critical

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